UNIVERSITY OF MINNESOTA

This is to certify that I have examined this bound copy of a doctoral thesis by

Philip Lloyd Osterlund

and have found it is complete and satisfactory in all respects,

and that any and all revisions required by the final examining committee have been made.

Name of Faculty Adviser

Signature of Faculty Adviser

Date GRADUATE SCHOOL Tensor Decompositions of the Regular Representation

of p-Groups Over Fields of Characteristic p

A Thesis

Submitted to the Faculty of The Graduate School

of The University of Minnesota By

Philip Lloyd Osterlund

In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

December, 1997 Chapter 0: Introduction

In this thesis we investigate certain properties of group rings. We describe a procedure for constructing and classifying all uniserial modules for the group ring of a p-group over a field of characteristic p. We also discuss non-trivial decompositions of the regular representation of a group ring into a tensor product of modules. We begin by explaining the context in which we will work, and we describe the less familiar notation.

Throughout this work, p is a prime and 𝔽 is a field of characteristic p. Unless stated otherwise, G is a p-group, and we study the group ring 𝔽G and its modules. We will almost always consider left modules. The group ring 𝔽G acts on itself as a module by left multiplication. This module is the regular representation of 𝔽G, which we also denote 𝔽G.

For a subgroup H of G, we will often wish to refer to the sum of the elements of H in 𝔽G, and we denote this by ∥H∥. The sum of the elements of a coset gH of H in G is denoted as g∥H∥ or ∥gH∥.

A group ring is a particular type of algebra. Let A be a finite dimensional algebra. A simple A-module S is a non-zero module which contains no submodules other than 0 and itself. A semisimple A-module is a module which is a direct sum of simple modules.

The radical of A, written Rad(A), is the intersection of all the maximal left (or right) ideals of A. If M is an A-module then we define Rad(M) = Rad(A)M. Powers of the radical are defined as Rad^{0}(M) = M, and Rad^{n}(M) = Rad(A) Rad^{n−1}(M) = Rad^{n}(A)M. The socle of M, written Soc(M), is the set of elements annihilated by Rad(A), i.e. Soc(M) = {m ∈ M | Rad(A)m = 0}. Powers of the socle are defined as Soc^{n}(M) = {m ∈ M | Rad^{n}(A)m = 0}.

The radical series of the module M is the series of modules,

M ⊇ Rad(M) ⊇ Rad^{2}(M) ⊇ · · · .

This is the fastest descending series of submodules with semisimple quotients. The socle series is described similarly, and is the fastest ascending series of submodules with semisimple quotients.

The Loewy length of a module M, written ℓℓ(M), is the minimal value r such that Rad^{r}(M) = 0, which is also equal to the minimal value r such that Soc^{r}(M) = M. Both the radical series and the socle series of M have length ℓℓ(M). The maximum possible Loewy length of an A-module is L = ℓℓ(A).

A uniserial module M is a module for which the following equivalent statements hold:

1) M has a unique composition series;

2) The submodules of M are totally ordered under inclusion;

3) The successive quotients of the radical series are simple or zero;

4) The successive quotients of the socle series are simple or zero. We study uniserial modules for several reasons. They have appeared widely in

the literature on the representation theory of finite-dimensional algebras over many years. See for example [F, B, ZH, AR&S]. In the context of group representation they have been studied in [Sh, LG], where they play an essential role in describing p-group structure. Also they were studied in [P].

In this thesis we show in Lemma 1.4 of Chapter 1 that each uniserial module is isomorphic to a cyclic submodule of the regular representation. This fact makes it possible to classify all uniserial modules up to isomorphism for any given finite dimensional group ring. Our work here can be viewed as treating a special case of open problems (1)-(4) on page 411 of [AR&S], and we present explicit classifications of uniserial modules for many groups of order 16. We are able to apply these results using the fact that uniserial modules are prime candidates for factors in a tensor decomposition of the regular representation of a group. We describe such decompositions explicitly in many cases.

In Chapter 1 we focus on classifying uniserial modules. In Section 1 we cover much of the theory behind the classification of uniserial modules. Some of the theory applies more broadly to cyclic modules. In Section 2 we outline a procedure for finding all uniserial submodules of a group ring. Section 3 consists of an example of the use of this procedure. We determine the set of all uniserial submodules of 𝔽Q_{8}, and classify all uniserial modules of 𝔽Q_{8} up to isomorphism.

Chapter 2 consists of calculation results, using the algorithm described in Chapter 1. Each section concentrates on a specific non-abelian group G of order 16. We restrict our attention to the groups that may not be written as the direct product of non-trivial subgroups of G. For each of these groups, we find the set of all uniserial submodules of 𝔽G, and classify all uniserial modules of 𝔽G, where 𝔽 is a field of characteristic 2.

In Chapter 3 we address the issue of tensor decompositions of the regular representation. We may take our motivation for the study of such decompositions in part from the simple observation for a direct product of groups that 𝔽[G×H] ≅ 𝔽G ⊗ 𝔽H. One may ask more generally when such a decomposition occurs, and this question is resolved by Carlson and Kovács [C&K] when G is an abelian pgroup and the decomposition is a tensor product of rings. Such decompositions also have relevance for the construction of multiple complexes, as considered by Benson and Carlson [B&C].

In Section 1 we develop some of the theory of tensor decompositions. In Section 2 we concentrate on permutation modules. This is another important class of modules which present themselves as factors in tensor decompositions. In Section 3, we use our results of Chapter 2 and the theory developed earlier in this chapter to determine all tensor decompositions of 𝔽G, where G is a group considered in Chapter 2.

We will make extensive use of the theory developed by Jennings [J] which we now summarize. Some of the notation is found in [Al 1].

For a p-group G, Jennings [J, Al 1, Sc] describes a decreasing series of subgroups, κ_{λ}(G) = {g ∈ G | g ≡ 1 modulo Rad^{λ}(𝔽G)}, which is sometimes called the Jennings series. This series of subgroups has the following properties:

1) [κ_{λ}, κ_{μ}] ⊆ κ_{λ+μ},

2) g^{p} ∈ κ_{ip} for all g ∈ κ_{i},

3) κ_{λ}/κ_{2λ} is elementary abelian. Furthermore, we may generate κ_{λ} as

κ_{λ} = ⟨[κ_{λ−1}, G], κ_{⌈λ/p⌉}^{(p)}⟩,

where κ_{1} = G, and ⌈λ/p⌉ is the least integer greater than or equal to λ/p, and κ_{λ}^{(p)} is the set of pth powers of elements of κ_{λ}. With this formulation, we have κ_{1}(G) = G, and it is clear that κ_{2}(G) is the Frattini subgroup of G.

For each i ≥ 1 let d_{i} be the dimension of κ_{i}/κ_{i+1}. Choose any elements x_{i,s} of G such that the set {x_{i,s}κ_{i+1} | 1 ≤ s ≤ d_{i}} forms a basis for κ_{i}/κ_{i+1}. Let x̄_{i,s} = x_{i,s} − 1 ∈ 𝔽G. There are |G| products of the form ∏ x̄_{i,s}^{α_{i,s}}, where the factors are listed in lexicographic order, and 0 ≤ α_{i,s} ≤ p − 1. The weight of such a product is defined to be ∑ iα_{i,s}. Jennings’ theorem states that the set of products of weight w lie in Rad^{w}(𝔽G), and forms a basis modulo Rad^{w+1}(G). Also, the set of products of weight ≥ w forms a basis for Rad^{w}(𝔽G).

Alperin comments [Al 1] that the order of the factors is irrelevant, as long as some order is chosen. After choosing a particular order for these factors, let {β_{i,t}} be the set of such products with weight i. Writing L = ℓℓ(𝔽G), the element β_{L−1,1} = ∏ x̄_{i,s}^{p−1} = ∥G∥ is a generator of the socle of 𝔽G, and has weight L − 1.

Often it is possible to choose an element x_{i,s} such that x_{i,s}^{p} is also one of our chosen coset representatives for κ_{ip+1} in κ_{ip}, and when this can be done we make this choice. As we do not need to choose the lexicographic ordering in the products, we choose an ordering that places x_{i,s} and x_{i,s}^{p} together.

For example, in the group D_{8} = ⟨x, y|x^{4} = y^{2} = 1⟩ we have κ_{1} = D_{8}, κ_{2} = ⟨x^{2}⟩, and κ_{3} = 1. We choose x_{1,1} = x, x_{1,2} = y, and x_{2,1} = x^{2}. We choose to write the factors in the order, x̄^{α_{1,1}}x̄^{2α_{2,1}}ȳ^{α_{1,2}} = x̄^{α_{1,1}+2α_{2,1}}ȳ^{α_{1,2}}, where 0 ≤ α_{i,s} < 2. The elements are then of the form x̄^{i}ȳ^{j} with 0 ≤ i < 4, and 0 ≤ j < 2.

Chapter 1: Classifying Uniserial Modules

In this chapter, after a few preliminary lemmas, we describe a procedure for classifying all uniserial modules of the group ring for a p-group. We then present an example to illustrate the use of this procedure.

§1 Uniserial Modules

The first four lemmas are found in the literature. See, for example, [Al 2], [H], and [AR&S].

We use the augmentation map of a group ring, which is the map, ϵ : 𝔽G → 𝔽 given by

ϵ(∑_{g∈G} ga_{g}) = ∑_{g∈G} a_{g},

where a_{g} is an element of the field 𝔽 for each element g of G. The augmentation ideal IG of 𝔽G is the kernel of this map. This ideal has codimension 1, and has the set {g − 1 | g ∈ G, g ≠ 1} as a basis. This set together with the element 1 forms a basis for 𝔽G.

Lemma 1.1. If the group G is generated by {x_{1}, …, x_{n}}, then the augmentation ideal is the two-sided ideal IG of the ring 𝔽G generated by the elements {x̄_{1}, …, x̄_{n}}. Thus we may write, IG = 𝔽Gx̄_{1} + ⋯ + 𝔽Gx̄_{n}.

Lemma 1.2. If G is a finite p-group, and 𝔽 is a field of characteristic p, then i) the trivial module is the only simple module of 𝔽G; ii) the radical and socle series of 𝔽G coincide; iii) the radical of 𝔽G is the augmentation ideal, IG; iv) the unique indecomposable projective 𝔽G-module is 𝔽G;

v) the unique indecomposable injective 𝔽G-module is 𝔽G.

Lemma 1.3. (Nakayama) If J is an ideal in a ring R with identity, then the following are equivalent.

i) J is contained in the Jacobson radical of R; ii) 1_{R} − j is a unit for every j ∈ J iii) If A is a finitely generated R-module such that JA = A, then A = 0; iv) If B is a submodule of a finitely generated R-module A such that A = JA+B, then A = B.

We relate this lemma to our situation where G is a p-group by noting that the Jacobson radical of the ring 𝔽G is the augmentation ideal, IG.

Lemma 1.4. A uniserial module M is a module for which the following equivalent statements hold:

i) M has a unique composition series; ii) The submodules of M are totally ordered under inclusion; iii) The successive quotients of the radical series are simple or zero; iv) The successive quotients of the socle series are simple or zero.

Lemma 1.5. If M is a uniserial module for 𝔽G, and G is a finite group, then i) M is cyclic (i.e. it can be generated by a single element); ii) M is a homomorphic image of 𝔽G; iii) M is isomorphic to a submodule of 𝔽G.

Proof. Uniserial modules have the unique composition series

M = Rad^{0}(M) ⊃ Rad(M) ⊃ … ⊃ Rad^{L}(M) = 0.

An element of M must therefore generate a power of the radical as a left module. If this element is not an element of the radical, then it must generate the module M.

Any module of an algebra A that is generated by a single element is a homomorphic image of A.

The module M has a simple socle, S = Soc(M). The module S is isomorphic to a submodule of 𝔽G. As 𝔽G is injective, the inclusion map of S into 𝔽G may be extended to a map ϕ on the module M. If the kernel of ϕ were non-zero, it would intersect the socle of M non-trivially. But ϕ is an extension of an injection of S, so the kernel must be 0. Thus ϕ is one-to-one, and M is isomorphic to its image in 𝔽G.

As a consequence, we will often assume that a uniserial module is a submodule of 𝔽G, and we designate this uniserial module as the module 𝔽GA, where A is an element of 𝔽G that generates the module. The zero module is considered uniserial with Loewy length 0.

Lemma 1.6. A non-zero module M is uniserial if and only if Rad(M) is uniserial and M/ Rad(M) is simple.

Proof. Let R = Rad(M). If M is a uniserial module, then M/R is simple. As R is a submodule of M, it too must have a unique composition series, and must also be uniserial. Conversely, if R is uniserial, and M/R is simple, then the successive quotients of the radical series of M are simple, so M is uniserial.

Corollary 1.7. Let 𝔽 be a field of characteristic p and let G be a finite p-group generated by the elements {x_{1}, x_{2}, …, x_{n}}. A non-zero 𝔽G-module M is uniserial if and only if M has a uniserial submodule N with dim(N) = dim(M) − 1 and 𝔽Gx̄_{i}M = N for some i.

Proof. In this case, all simple modules of 𝔽G are isomorphic to 𝔽. If M is a non-zero 𝔽G-module, then

Rad(M) = Rad(𝔽G)M = IGM = (𝔽Gx̄_{1} + ⋯ + 𝔽Gx̄_{n})M.

If M is uniserial, then N = Rad(M) must also be uniserial, and M/N must be simple. The modules 𝔽Gx̄_{j}M are all submodules of Rad(M), and generate Rad(M). Since the submodules of M are totally ordered under inclusion, there must be an index i such that 𝔽Gx̄_{i}M contains the modules 𝔽Gx̄_{j}M for all j. As Rad(M) = IGM, we must have N = Rad(M) = 𝔽Gx̄_{i}M. Since dim(M/N) = dim(𝔽) = 1, we have dim(N) = dim(M) − 1.

Conversely, if N = 𝔽Gx̄_{i}M is a uniserial module, and dim(N) = dim(M) − 1, then clearly N ⊆ Rad(M), as x̄ ∈ Rad(𝔽G). We have dim(M/N) = 1, so the quotient is simple. Thus, by the lemma, M is uniserial.

In the following proposition we regard Hom_{𝔽G}(𝔽GA, 𝔽G) as a right 𝔽Gmodule by means of the action (ϕ · C)(z) = ϕ(z) · C, where C ∈ 𝔽G, and z ∈ 𝔽G.

Proposition 1.8. If A is an element of 𝔽G then Hom_{𝔽G}(𝔽GA, 𝔽G) ≅ A𝔽G as a right 𝔽G-module.

Proof. Let ι : 𝔽GA → 𝔽G be the inclusion map. Given an 𝔽G homomorphism ϕ : 𝔽GA → 𝔽G, by the injectivity of 𝔽G, there is an extension ψ : 𝔽G → 𝔽G such that ϕ = ψ ∘ ι. Let B = ψ(1). Given an element x ∈ 𝔽G, we have ψ(x) = xψ(1) = xB, as ψ is an 𝔽G-homomorphism. Thus the homomorphism acts as right multiplication by the element B. Restricting this action to 𝔽GA, we have ϕ(xA) = ψι(xA) = xAB. If ψ′ is another extension of ϕ, we have ϕ(x) = xAψ′(1), so Aψ(1) = Aψ′(1) = AB. Thus we associate the homomorphism ϕ with the element AB. We designate this association by writing ϕ = ϕ_{A,B}.

Given an element AC of A𝔽G, let ϕ_{A,C} : 𝔽GA → 𝔽G be the map given by ϕ_{A,C}(xA) = xAC. This is an 𝔽G-homomorphism on 𝔽GA, and ϕ_{A,B} = ϕ_{A,C} if and only if AB = AC. When adding homomorphisms we have

\begin{aligned}(ϕ_{A,B} + ϕ_{A,C})(xA) & = xAB + xAC = xA(B + C) = ϕ_{A,B+C}(x), so \\ ϕ_{A,B} + ϕ_{A,C} & = ϕ_{A,B+C}. \end{aligned}

Also, ϕ_{A,B}(x) · C = xABC, so ϕ_{A,B} · C = ϕ_{A,BC}.

We have shown that there is a one-to-one correspondence between elements AB of A𝔽G and homomorphisms ϕ_{A,B} : 𝔽GA → 𝔽G, and that both sets act as right 𝔽G-modules in the same manner.

Corollary 1.9. Let G be a p-group and let A and B be elements of 𝔽G. Then 𝔽GA ≅ 𝔽GB if and only if 𝔽GB = 𝔽GAx, where x ∈ 𝔽G is some unit.

Proof. Assume that 𝔽GA ≅ 𝔽GB. The module 𝔽GB is a submodule of 𝔽G, so by Proposition 1.8 we know that there is an isomorphism ϕ_{A,x} : 𝔽GA → 𝔽GB whose image is 𝔽GAx = 𝔽GB. The element x is not an element of Rad(𝔽G) because the Loewy lengths of 𝔽GA and 𝔽GB must be the same and the Loewy length of 𝔽GAx is determined by the power of Rad(𝔽G) in which Ax lies. All elements of 𝔽G which are not elements of the radical are units, so x is a unit.

Conversely, assume that x is a unit of 𝔽G, and that 𝔽GB = 𝔽GAx. Then by Proposition 1.8, 𝔽GB is a homomorphic image of 𝔽GA. If we apply the map ϕ_{B,x^{−1}} to the module 𝔽GB, we see that 𝔽GA = 𝔽GBx^{−1}, so 𝔽GA is also a homomorphic image of 𝔽GB, so the modules are isomorphic.

We now come to our first substantial result.

Theorem 1.10. Let G be a p-group. If A ∈ 𝔽G, and the left 𝔽G-module 𝔽GA is uniserial of length l, then the right 𝔽G-module A𝔽G is also uniserial of length l.

Proof. If 𝔽GA is of length 1, then 𝔽GA is the trivial module, so A is a multiple of ∥G∥. The group ring element ∥G∥ is central, so 𝔽GA = 𝔽G∥G∥ = ∥G∥𝔽G = A𝔽G, and is uniserial of length 1.

Assume the statement of the theorem is true for all j < l. Then if A is an element of 𝔽G which generates a left 𝔽G-module of length j < l, then Hom(𝔽GA, 𝔽G) ≅ A𝔽G, and by induction, these have dimension j.

Let A^{(l)} be an element of 𝔽G such that 𝔽GA^{(l)} is uniserial of length l. Since 𝔽G is injective and contains every simple module in its socle, every indecomposable left 𝔽G-module whose socle is simple is isomorphic to a submodule of 𝔽G. Thus there is a submodule M of 𝔽G that is isomorphic to 𝔽GA^{(l)}/ Soc(𝔽GA^{(l)}). Let ϕ be the composite homomorphism 𝔽GA^{(l)} → 𝔽GA^{(l)}/ Soc(𝔽GA^{(l)}) → M. Then ϕ corresponds to an element A^{(l)}B ∈ A^{(l)}𝔽G, for some element B of 𝔽G, and ϕ has the form ϕ(gA^{(l)}) = gA^{(l)}B. Thus M = 𝔽GA^{(l)}B, and it is uniserial of length l − 1. By our assumption, A^{(l)}B𝔽G is a uniserial right 𝔽G-module of length l − 1.

Given C ∈ 𝔽G, let u = ϵ(C) ∈ 𝔽, and r = C − u ∈ Rad(𝔽G). Then 𝔽GA^{(l)}r is a homomorphic image of 𝔽GA^{(l)}. Since A^{(l)}r is in a higher power of the radical than A^{(l)}, it is in a lower power of the socle of 𝔽G (since the radical and socle series of 𝔽G coincide when G is a p-group), and it must be true that ℓℓ(𝔽GA^{(l)}r) < ℓℓ(𝔽GA^{(l)}). This implies that 𝔽GA^{(l)}r has dimension less than l. This means that it is also a homomorphic image of 𝔽GA^{(l)}B, so A^{(l)}r ∈ A^{(l)}B𝔽G. The element A^{(l)}C = A^{(l)}u + A^{(l)}r lies in 𝔽A^{(l)} + A^{(l)}B𝔽G, since u is an element of 𝔽. Since C was an arbitrary element of 𝔽G he have A^{(l)}𝔽G = 𝔽A^{(l)} + A^{(l)}B𝔽G, and it has dimension l. Since A^{(l)} ∈ Soc^{l}(𝔽G), it must be true that ℓℓ(A^{(l)}𝔽G) = l. This means that A^{(l)}𝔽G is a uniserial module of length l.

We have shown that the set of homomorphisms from a uniserial left 𝔽Gmodule into 𝔽G forms a uniserial right 𝔽G-module of the same length.

Lemma 1.11. Let G be a p-group and let A and B be two non-zero elements of 𝔽G. The cyclic modules 𝔽GA and 𝔽GB are equal if and only if B = fA + r for some non-zero element f of 𝔽 and some element r of Rad(𝔽GA).

Proof. Assume 𝔽GA = 𝔽GB. As B ∈ 𝔽GA, we may write B = xA, for some element x of 𝔽G. The radical of 𝔽G is a subspace of codimension 1, so we may write x = f1 + s, for some element f of 𝔽, and some element s of Rad(𝔽G). Thus B = (f1 + s)A = fA + r, where r = sA ∈ Rad(𝔽GA). The element f must be non-zero because 𝔽Gr ⊆ Rad(𝔽GA) ≠ 𝔽GA.

Assume that B = fA + r, for some non-zero element f ∈ 𝔽 and some element r ∈ Rad(𝔽GA). Then A ∈ 𝔽GB + Rad(𝔽GA) so 𝔽G = 𝔽GB + Rad(𝔽GA). Therefore by Nakayama’s lemma 𝔽GA = 𝔽GB.

Lemma 1.12. Let G be a p-group. If A and B are elements of 𝔽G, λ = ℓℓ(𝔽GA), and 𝔽GA ≅ 𝔽GB, then B = fA + r for some non-zero element f of 𝔽 and some element r ∈ Soc^{λ−1}(𝔽G).

Proof. Assume that 𝔽GA ≅ 𝔽GB. By Corollary 1.9 we know that 𝔽GB = 𝔽GAx for some x ∈ 𝔽G. We rewrite x = f_{1} + s_{1}, where 0 ≠ f_{1} ∈ 𝔽 and s_{1} ∈ Rad(𝔽G). Then 𝔽GB = 𝔽GA(f_{1} + s_{1}). Since f_{1} ≠ 0, we have Af_{1} ∈ Soc^{λ}(𝔽G) − Soc^{λ−1}(𝔽G). As s_{1} ∈ Rad(𝔽G), and Soc^{λ}(𝔽G) is a two-sided ideal, we have As_{1} ∈ Soc^{λ−1}(𝔽G). By the previous lemma, we must have a non-zero element f_{2} ∈ 𝔽 and an element r_{2} ∈ Rad(𝔽GA(f_{1} +s_{1})) such that B = f_{2}A(f_{1} +s_{1})+r_{2} = fA+r, where f = f_{2}f_{1} and r = f_{2}As_{1}+r_{2}. As A ⊆ Soc^{λ}(𝔽G), we have r ∈ Soc^{λ−1}(𝔽G).

We use these results to justify the algorithm described in the next section.

§2 A Procedure for Finding Uniserial Modules

We are about to describe an algorithm for finding uniserial modules of 𝔽G. The algorithm works when G is a finite p-group and 𝔽 is a field of characteristic p, and we make these assumptions. We choose a set of generators of G, {x_{1}, …, x_{n}}.

Algorithm 2.1. Suppose we are given a uniserial module N ⊆ 𝔽G of length l − 1. We construct all uniserial submodules of 𝔽G having length l which contain N as a submodule by the following procedure:

We find all elements A ∈ 𝔽G with the properties

i) x̄_{i}A ∈ N, ∀i ii) for some i, x̄_{i}A generates N.

Then the module 𝔽GA is uniserial of the kind specified. Every uniserial module may be constructed in this way.

Remark 2.2. Such A must necessarily lie in Soc^{l}(𝔽G). If we start with the submodule 0 of 𝔽G, we generate all uniserial submodules of 𝔽G by successive use of Algorithm 2.1. Since every uniserial module is isomorphic to a submodule of 𝔽G, this algorithm produces all isomorphism types of uniserial modules of 𝔽G. Note, however, that distinct uniserial submodules of 𝔽G may in fact be isomorphic. We explain later how to distinguish between isomorphism classes of uniserial modules.

Let A be an element of the kind specified in the algorithm, and let M = 𝔽GA. Then if 0 ≠ f ∈ 𝔽, and r ∈ Rad(M) = N, then fA + r also generates M. We would like to find one distinguished generating element for each module. We may then classify all uniserial modules of 𝔽G having length l by the set of distinguished generating elements.

A description of the distinguished generating element of each module will now be given.

Let B = {b_{i,λ}} be a basis of 𝔽G such that B_{λ} = {b_{i,μ}|μ ≤ λ} is a basis for Soc^{λ}(𝔽G). We will call such a basis a filtered basis of 𝔽G. Among the basis elements with a common second subscript, set up an ordering that is reflected in the first subscript. Write elements of 𝔽G as linear combinations of these basis elements.

Lemma 2.3. If M is a uniserial submodule of 𝔽G with Loewy length l, then there are unique sets {A^{(λ)}|1 ≤ λ ≤ l} ⊆ M and {b_{i_{λ},λ}|1 ≤ λ ≤ l} ⊆ B such that

i) Soc^{λ}(M) = 𝔽GA^{(λ)} ii) i_{λ} is the minimal i such that the coefficient of b_{i,λ} is non-zero in A^{(λ)} iii) The coefficient of b_{i_{λ},λ} in A^{(λ)} is 1 iv) The coefficient of b_{i_{λ},λ} in A^{(μ)} is 0 for all μ > λ.

Proof. If l = 1, then the only choice is A^{(1)} = b_{1,1} = ||G||. We use induction on the Loewy length of M. Assume the statement is true for N = Rad(M) ≠ 0. The unique sets {A^{(λ)} | 1 ≤ λ < l} and {b_{i_{λ},λ} | 1 ≤ λ < l} associated with N then exist. Let A be a generator of M. As M is uniserial, M/N is isomorphic to the trivial module. The choice of b_{i_{l},l} must therefore be unique, regardless of the generator A for M. Let A_{1} = A/c, where c is the coefficient of b_{i_{l},l} in A. Let c_{λ} be the coefficient of b_{i_{λ},λ} in A_{1}, where 1 ≤ λ < l. Let

A^{(l)} = A_{1} − ∑_{λ=1}^{l−1} c_{λ}A^{(λ)}.

Then the statement holds for M as well. The construction of b_{i_{l},l} and A^{(l)} did not depend on the choice of A.

In the algorithm for finding all uniserial modules, Lemma 2.3 may be used to limit the search for generating elements.

As we are dealing with a p-group, the radical and socle series of 𝔽G coincide, or more precisely, Rad^{L−λ}(𝔽G) = Soc^{λ}(𝔽G), where L = ℓℓ(𝔽G). Thus, it is convenient to use Jennings’ construction of a filtered basis, filtered with respect to powers of the radical.

A note on extending the procedure to group rings which have more than one simple module: finding a series of distinguished generating elements for a uniserial module and its submodules is a bit more cumbersome. One must deal with Gorbits of the generators when choosing b_{λ,i_{λ}}, among other details.

§3 An Example of the Procedure for Finding Uniserial Submodules of 𝔽G

As a first example of the procedure, let

G = Q_{8} = ⟨x, y|x^{4} = 1, x^{2} = y^{2}, ^{y}x = x^{−1}⟩,

and let 𝔽 be a field of characteristic 2. The Jennings series of subgroups is κ_{1} = Q_{8} ⊃ κ_{2} = ⟨x^{2}⟩ ⊃ κ_{3} = ⟨1⟩. By Jennings’ theorem, we may choose generating sets {x_{1,1} = x̄, x_{1,2} = ȳ}, and {x_{2,1} = x̄^{2}}. Since x̄^{2} = ȳ^{2}, we may use x̄^{2}ȳ for a basis element, going against Jennings’ specifications, and still be using the same elements.

We obtain the following filtered basis for 𝔽G:

β_{0,1} = 1
β_{1,1} = ȳ  β_{1,2} = x̄
β_{2,1} = x̄ȳ  β_{2,2} = x̄^{2}
β_{3,1} = x̄^{2}ȳ  β_{3,2} = x̄^{3}
β_{4,1} = x̄^{3}ȳ

The notation we wish to use for our basis is b_{λ,i} = β_{(5−λ),i}, emphasizing powers of the socle, rather than powers of the radical.

The geometric presentation of the basis elements will be exploited to represent elements of 𝔽G. A vector in 𝔽G is written with the shape given above, with the coefficient for each basis element in the corresponding location. This gives a clearer meaning than the standard linear way to explicitly express a vector.

From the previous section, we let A^{(1)} = b_{1,1}.

Since Soc^{2}(𝔽G)/ Soc(𝔽G) is semisimple, and Rad(Soc^{2}(𝔽G)) is one dimensional, any element of Soc^{2}(𝔽G) − Soc(𝔽G) generates a 2-dimensional uniserial module. The set of distinguishing generators for 2-dimensional uniserial submodules is {A_{r}^{(2)} = b_{2,1} + rb_{2,2} | r ∈ 𝔽} ∪ {B^{(2)} = b_{2,2}}.

It is necessary to take into account the left action of 𝔽Q_{8}. As x and y generate G, the radical of 𝔽G is generated by x̄ and ȳ. If M is a module, Rad(M) = Rad(𝔽G)M = Span(x̄M, ȳM). We need to find the left actions of x̄ and ȳ on the basis elements.

The left action of x̄ on the basis elements is clear. Remember that x̄^{4} = (x − 1)^{4} = x^{4} − 1 = 0, as we are dealing with a field of characteristic 2. It may be calculated that ȳx̄ = x̄ȳ + (x̄^{2} + x̄^{3})(ȳ + 1), and that ȳ commutes with x̄^{2} and x̄^{3}. Also, ȳ^{2} = x̄^{2}. Thus ȳ x̄ȳ = x̄^{2}ȳ + x̄^{3} + x̄^{3}ȳ.

In vector notation, as described above, we write an arbitrary element A of 𝔽Q_{8}, as well as x̄A, and ȳA as

A = \begin{aligned}& a_{5} \\ a_{4,1} & & a_{4,2} \\ a_{3,1} & & a_{3,2} \\ a_{2,1} & & a_{2,2} \\ & a_{1} \end{aligned} ,  x̄A = \begin{aligned}& 0 \\ 0 & & a_{5} \\ a_{4,1} & & a_{4,2} \\ a_{3,1} & & a_{3,2} \\ & a_{2,1} \end{aligned} ,

ȳA = \begin{aligned}& 0 \\ a_{5} & & 0 \\ a_{4,2} & & a_{4,1} + a_{4,2} \\ a_{4,2} + a_{3,1} + a_{3,2} & & a_{4,2} + a_{3,1} \\ & a_{4,2} + a_{3,1} + a_{2,2} \end{aligned}

or, more conveniently,

A = \begin{aligned}& a \\ b & & c \\ d & & e \\ f & & g \\ & h \end{aligned} ,  x̄A = \begin{aligned}& · \\ · & & a \\ b & & c \\ d & & e \\ & f \end{aligned} ,  ȳA = \begin{aligned}& · \\ a & & · \\ c & & b + c \\ c + d + e & & c + d \\ & c + d + g \end{aligned}

where the character “·” deemphasizes the zero entries.

When multiplying on the right we obtain

Ax̄ = \begin{aligned}& · \\ · & & a \\ b & & b + c \\ b + d & & b + d + e \\ & b + d + f \end{aligned} , and Aȳ = \begin{aligned}& · \\ a & & · \\ c & & b \\ e & & d \\ & g \end{aligned}

Using our vector notation, the distinguished generators for two-dimensional uniserial modules are:

A_{r}^{(2)} = \begin{matrix}& · \\ · & & · \\ · & & · \\ 1 & & r \\ & · \end{matrix} (r ∈ 𝔽), B^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & 1 \\ & · \end{aligned}

Assume A is a distinguished generator for a uniserial module of length 3. We must have a = b = c = 0, since A ∈ Soc^{3}(𝔽G). These restrictions give

A = \begin{aligned}& · \\ · & & · \\ d & & e \\ f & & g \\ & h \end{aligned} ,  x̄A = \begin{aligned}& · \\ · & & · \\ · & & · \\ d & & e \\ & f \end{aligned} ,  ȳA = \begin{aligned}& · \\ · & & · \\ · & & · \\ d + e & & d \\ & d + g \end{aligned}

Either x̄A or ȳA must be an element of Soc^{2}(𝔽G)−Soc(𝔽G), so d or e must be non-zero. This forces the coefficient of b_{2,1} and b_{2,2} to be non-zero in at least one of x̄A and ȳA. As they must both be in the same module, both coefficients must be non-zero. We must have an extension of 𝔽GA_{r}^{(2)}, where r ≠ 0. By Lemma 2.3 parts ii) and iii), we need d = 1, which forces e = r, and 1 + r = 1/r. This implies that r must be a primitive third root of unity, as char(𝔽) = 2. By Lemma 2.3iv), we may assume that f = h = 0. We then have,

A_{r,s}^{(3)} = \begin{aligned}& · \\ · & & · \\ 1 & & r \\ · & & s \\ & · \end{aligned} ,

x̄A_{r,s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ 1 & & r \\ & · \end{aligned} = A_{r}^{(2)},  ȳA_{r,s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ 1 + r & & 1 \\ & 1 + s \end{aligned} = \frac{1}{r}A_{r}^{(2)} + (1 + s)A^{(1)}

where r is a primitive third root of unity and s ∈ 𝔽.

Now we show that there is no uniserial module in 𝔽G of length 4.

Assume A is a distinguished generator for a uniserial module of length 4. By the multiplication shown above, we must have a = 0. The element x̄A must have a non-zero coefficient for either b_{3,1} or b_{3,2}. Since this must be a generator of a uniserial module of length 3, both coefficients must be non-zero, so both b and c must be non-zero. Using Lemma 2.3 parts iii) and iv), we may assume that b = 1 and d = f = h = 0. These restrictions give

A = \begin{aligned}& · \\ 1 & & c \\ · & & e \\ · & & g \\ & · \end{aligned} ,  x̄A = \begin{aligned}& · \\ · & & · \\ 1 & & c \\ · & & e \\ & · \end{aligned} ,  ȳA = \begin{aligned}& · \\ · & & · \\ c & & 1 + c \\ c + e & & c \\ & c + g \end{aligned}

We must choose c = r and e = s, giving

A = \begin{aligned}& · \\ 1 & & r \\ · & & s \\ · & & g \\ & · \end{aligned} ,  x̄A = \begin{aligned}& · \\ · & & · \\ 1 & & r \\ · & & s \\ & · \end{aligned} ,  ȳA = \begin{aligned}& · \\ · & & · \\ r & & 1 + r \\ r + s & & r \\ & r + g \end{aligned}

We now have x̄A = A_{r,s}^{(3)}. For ȳA to be in 𝔽GA_{r,s}^{(3)}, we must have

ȳA = rA_{r,s}^{(3)} + (r + s)A_{r}^{(2)} + (r + g)A^{(1)} = \begin{aligned}& · \\ · & & · \\ r & & r^{2} \\ r + s & & rs + r^{2} + rs \\ & r + g \end{aligned}

In our field, rs + r^{2} + rs = r^{2} = r + 1. However, the coefficient for b_{2,2} needs to be r, not r + 1, which leads to a contradiction. This leads to the following theorem.

Theorem 3.1. The isomorphism classes of uniserial modules of 𝔽Q_{8} are in 1-to-1 correspondence with the distinguished generators

{A^{(1)}, A_{r}^{(2)}, B^{(2)}, A_{ω,0}^{(3)} | r, ω ∈ 𝔽, ω^{2} + ω + 1 = 0}.

(If 𝔽 does not contain a primitive third root of unity, then there is no uniserial module of length 3 for 𝔽Q_{8}.)

Proof. By Lemma 1.5 we know that all uniserial left modules of 𝔽Q_{8} are isomorphic to submodules of 𝔽Q_{8}, taken as a left module. By Lemma 2.3, and the work above, all uniserial submodules of 𝔽Q_{8} may be generated by one of the elements in the list {A^{(1)}, A_{r}^{(2)}, B^{(2)}, A_{ω,s}^{(3)} | r, s, ω ∈ 𝔽, ω^{2} + ω + 1 = 0}. The action of 𝔽Q_{8} on a module characterizes the module.

There is only one isomorphism class of 1-dimensional modules for 𝔽Q_{8}. They are thus isomorphic to 𝔽Q_{8}A^{(1)}.

Assume M is a 2-dimensional uniserial module generated by A. If x̄A = 0, then x̄M = 0, and M ≅ 𝔽GB^{(2)}. If x̄A ≠ 0, then ȳA = rx̄A, for some r ∈ 𝔽. This only occurs when M ≅ 𝔽GA_{r}^{(2)}.

Two modules M and N are isomorphic over 𝔽G if and only if the group representations of G with respect to M and N are isomorphic.

Assume M is a 3-dimensional uniserial module. For some ω, s ∈ 𝔽, M = 𝔽GA_{ω,s}^{(3)}. Let A = A_{ω,s}^{(3)} − (s + 1)A_{ω}^{(2)}. As this element has a non-zero coefficient for A_{ω,s}^{(3)}, it generates M. A basis for M is given by 𝔅 = {A, x̄A, x̄^{2}A}, where x̄A = A_{ω}^{(2)} − (s + 1)A^{(1)}, and x̄^{2}A = A^{(1)}. The action of ȳ on this basis is then ȳA = (1 + ω)A_{ω}^{(2)} + (1 + s)A^{(1)} − ω(s + 1)A^{(1)} = (1 + ω)x̄A, and ȳ x̄A = ωx̄^{2}A. Thus, regardless of s, there is a basis for M such that the action of x and y are given by the matrices

ρ(x) = ( \begin{matrix}1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{matrix} ) , ρ(y) = ( \begin{matrix}1 & 0 & 0 \\ 1 + ω & 1 & 0 \\ 0 & ω & 1 \end{matrix} )

as x = x̄ + 1, y = ȳ + 1. Because Rad(M) = 𝔽Q_{8}A_{ω}^{(2)} ≅ 𝔽̸Q_{8}A_{ω+1}^{(2)}, there are exactly two isomorphism classes of uniserial modules of length 3 for 𝔽Q_{8}.

We now present an alternative proof of Theorem 3.1 which uses a more practical method. We use this second method to find the isomorphism classes of uniserial modules in the next chapter.

Proof. By Lemma 1.5 we know that all uniserial left modules of 𝔽Q_{8} are isomorphic to submodules of 𝔽Q_{8}, taken as a left module. By Lemma 2.3, and the work above, all uniserial submodules of 𝔽Q_{8} may be generated by one of the elements in the list {A^{(1)}, A_{r}^{(2)}, B^{(2)}, A_{ω,s}^{(3)} | r, s, ω ∈ 𝔽, ω^{2} + ω + 1 = 0}. Let A be one of these generators, and let M = 𝔽Q_{8}A. By Proposition 1.8 we have Hom(𝔽Q_{8}A, 𝔽Q_{8}) ≅ A𝔽Q_{8}, so the set of homomorphisms is spanned by the set {Aβ_{λ,i}}. For each of these distinguished elements we have,

A1 = A,  A^{(n)}β_{λ,i} = 0  (λ > n)
A_{r}^{(2)}x̄ = A^{(1)},  A_{r}^{(2)}ȳ = rA^{(1)}
B^{(2)}x̄ = 0,  B^{(2)}ȳ = A^{(1)}
A_{ω,s}^{(3)}x̄ = A_{ω^{2}}^{(2)} + A^{(1)}, A_{ω,s}^{(3)}ȳ = ωA_{ω^{2}}^{(2)} + sA^{(1)},
A_{ω,s}^{(3)}x̄^{2} = A^{(1)},  A_{ω,s}^{(3)}x̄ȳ = ωA^{(1)}

From this information we may list all homomorphic images of each uniserial module.

There is only one isomorphism class of 1-dimensional modules for 𝔽Q_{8}. They are thus isomorphic to 𝔽Q_{8}A^{(1)}.

Let M_{r}^{(2)} = 𝔽GA_{r}^{(2)}. The homomorphic images of A_{r}^{(2)} are of the form αA_{r}^{(2)}+ βA^{(1)}, where α and β are elements of 𝔽. All such elements are elements of M_{r}^{(2)}. The isomorphisms occur precisely when α ≠ 0. Thus, for each r ∈ 𝔽, the module M_{r}^{(2)} forms its own isomorphism class. Likewise, N^{(2)} = 𝔽GB^{(2)} forms its own isomorphism class.

Let M_{ω,s}^{(3)} = 𝔽GA_{ω,s}^{(3)}. Given an 𝔽G-homomorphism ϕ of M, we wish to find the distinguished generating element of ϕ(M). We know that ϕ(A(3)_{ω,s}) is of the form αA_{ω,s}^{(3)} + βA_{ω^{2}}^{(2)} + γA^{(1)}. Writing this in our vector notation, this is

ϕ ( \begin{matrix}& · \\ · & & · \\ 1 & & ω \\ · & & s \\ & · \end{matrix} ) = \begin{aligned}& · \\ · & & · \\ α & & αω \\ β & & αs + βω^{2} \\ & γ \end{aligned}

If α = 0, then ϕ is not an isomorphism. By using different values for β and γ we see that M_{ω^{2}}^{(2)} is a homomorphic image of M_{ω,s}^{(3)}, as is M^{(1)} = 𝔽GA^{(1)}.

If α ≠ 0, the image of M_{ω,s}^{(3)} is 3-dimensional, so ϕ must be an isomorphism. We find the elements x̄^{2}/αϕ(A_{ω,s}^{(3)}) = A^{(1)} and (x̄ϕ(A_{ω,s}^{(3)}) − βA^{(1)})/α = A_{ω}^{(2)} are elements in the image of ϕ. Thus the element

\begin{aligned}· \\ · & & · \\ α & & αω \\ β & & αs + βω^{2} \\ & γ \end{aligned} − \begin{aligned}· \\ · & & · \\ · & & · \\ β & & βω \\ & γ \end{aligned} = \begin{aligned}· \\ · & & · \\ α & & αω \\ · & & αs + β \\ & · \end{aligned}

is also an element of ϕ(M_{ω,s}^{(3)}).

If we look at all possible isomorphisms of M_{ω,s}^{(3)}, we see that setting α = 1 and letting β vary over 𝔽 that M_{ω,s}^{(3)} ≅ M_{ω,s′}^{(3)} for all s′ ∈ 𝔽. We may choose the module M_{ω,0}^{(3)} to be the representative of this isomorphism class.

We now display two lattices of all the uniserial submodules of 𝔽Q_{8}. The first lattice shows the sub-module structure of the uniserial modules. A submodule of another module is listed below it, and there is a vertical or diagonal connection between them. The horizontal arrow indicates that the module M_{ω}^{(2)} is a module of the type M_{r}^{(2)}, where r ∈ 𝔽. We define the modules as we did in the proof.

[figure: M_{ω,s}^{(3)}; M_{r}^{(2)}; M_{ω}^{(2)}; N^{(2)}; M^{(1)}]

The second lattice describes the homomorphism structure of the modules. If M is a homomorphic image of N then the module M appears at a lower level in the lattice than the module N, and there is a vertical or diagonal line connecting them. The horizontal arrow indicates that the module M_{ω^{2}}^{(2)} is a module of the type M_{r}^{(2)}, where r ∈ 𝔽. Each isomorphism class is represented by a “bucket”. The information inside the bucket describes all modules that lie in the same isomorphism class. The information outside the bucket describes all isomorphism classes of this type. Thus the top bucket implies that M_{ω,s}^{(3)} ≅ M_{ω′,s′}^{(3)} whenever s and s′ are elements of 𝔽, and only when ω = ω′. These modules exist for all ω ∈ 𝔽 such that ω^{2} + ω = 1.

[figure: \underline{M_{ω,s}^{(3)}_{s∈𝔽}}_{ω^{2}+ω=1}; \underline{M_{r}^{(2)}}_{r∈𝔽}; \underline{M_{ω^{2}}^{(2)}}_{ω^{2}+ω=1}; \underline{N^{(2)}}; \underline{M^{(1)}}]

In the next chapter we use similar notation. For simplicity, we place the name of the generator of a uniserial module in the bucket, rather than the name of the module. In doing this, we do not need to name all the modules, and there is less possibility of confusing the generators of the modules.

Chapter 2: Classification of Uniserial Modules for Groups of Order 16

In this chapter we explicitly describe the uniserial 𝔽G-modules, where 𝔽 is a field of characteristic 2, and G is one of a certain collection of 2-groups. We restrict ourselves to the groups of order 16 that are non-abelian, and may not be written as the direct product of two non-trivial groups. There are seven such groups ([H&S, Sc, As]), and we denote these G_{1}, (2 × 4).2, Mod_{16}, D_{16}, SD_{16}, Q_{16}, and D_{8}Y4. Each group will be considered in its own section.

Each section concentrates on one specific group, G. In each section the presentation of material follows this pattern:

• A presentation of G is given.

• We describe the partially ordered set of all uniserial submodules of the regular representation 𝔽G, were the ordering is by inclusion.

• We describe the partially ordered set of all isomorphism classes of 𝔽G modules,

where M ≤ N indicates that M is a homomorphic image of N. The modules are then defined explicitly in the following way:

• A basis is given for the regular representation 𝔽G.

• An arbitrary element A of 𝔽G is given.

• We give actions of generators of 𝔽G on A.

• We list the uniserial submodules U of 𝔽G in increasing order of dimension. We do this for each uniserial submodule by listing a distinguished generator B of U, along with the left and right action of generators of 𝔽G on B. Each module is denoted by listing its distinguished generator. These genera-

tors are written as symbols such as A_{r,s}^{(3)}. Here the letter A denotes the family to which the module belongs, the superscript denotes the dimension of the module generated by this element, and the subscripts are parameters which lie in 𝔽. If different values are given to these subscripts, then different modules are represented. The particular definition of each generator is given after the diagrams of inclusion and homomorphism.

In the diagram that represents the partially ordered set of uniserial modules, if for example, A_{r,s}^{(3)} is above A_{r}^{(2)} with a solid line connecting them, this indicates that, for any fixed r ∈ 𝔽 and all s ∈ 𝔽, the module generated by A_{r,s}^{(3)} contains the module generated by A_{r}^{(2)}. In the diagrams, if there is an arrow from, let us say, A_{1}^{(2)} to A_{r}^{(2)}, this indicates that the module generated by A_{1}^{(2)} is a specific case of the more generally described module generated by A_{r}^{(2)}. A dotted line between C_{s,t}^{(3)} and A_{r,s}^{(3)} indicates that there is overlap in the definitions of the modules generated by these generators. The overlap is described with the dotted lines.

In the homomorphism diagram, isomorphism classes are represented by buckets. Each bucket contains a list of all items which are in a particular isomorphism class. We describe the general bucket by analogy with the description of the following specific bucket:

\underline{A_{r,s,t}^{(4)}_{s∈𝔽}} ; r ≠ 1

All information pertaining to a specific isomorphism class is contained inside of the bucket. All information listed outside of the bucket is used to distinguish different isomorphism classes, or to determine permitted values of variables. Given r, s, t ∈ 𝔽, where r ≠ 1, our sample bucket represents the isomorphism class of 𝔽GA_{r,s,t}^{(4)}. The variable s is permitted to vary over all the elements of 𝔽. Thus if r ≠ 1, then 𝔽GA_{r,s,t}^{(4)} ≅ 𝔽GA_{ρ,σ,τ}^{(4)} if and only if ρ = r and τ = t, where all parameters are elements of 𝔽.

Assume, for example, that in a diagram the bucket containing A_{r,s,t}^{(4)} is above another bucket containing C_{r,s}^{(3)}. This indicates that there is a surjective homomorphism from the module generated by A_{r,s,t}^{(4)} to the module generated by C_{r,s}^{(3)}. It must be noted that r and s have the same values in each of these generators.

The homomorphism diagram lists all situations where there is a homomorphism from one uniserial module onto another uniserial module. If two modules are isomorphic, then their distinguished generators are listed in the same bucket. If there is a module homomorphism ϕ : M → N, then the bucket containing the distinguished generator of M is above the bucket containing the distinguished generator of N, and there is a line or series of lines connecting them. There may also be parameters in each bucket, and these must match.

§1 Uniserial submodules of 𝔽G_{1}

Group: G_{1} = ⟨x, y | x^{4} = y^{4} = 1, yx = x^{−1}y^{−1}, {x^{2}, y^{2}} ⊂ Z⟩, where Z is the center of the group.

This is the partially ordered set of uniserial 𝔽G_{1} submodules:

[figure: D_{s,t,u}^{(4)}; C_{s,t,u,v}^{(4)}; s=1,u=0; r=1; A_{r,t,v}^{(4)}; B_{s,t}^{(4)}; D_{s}^{(3)}; C_{s,t}^{(3)}; s=1; r=1; A_{r,t}^{(3)}; B_{s}^{(3)}; A_{1}^{(2)}; A_{r}^{(2)}; B^{(2)}; A^{(1)}]

This is the homomorphism diagram for uniserial 𝔽G_{1} modules:

[figure: \underline{A_{r+1,s,t}^{(4)}_{rs+rt+s^{2}=α}}_{r,α∈𝔽;r≠0}; \underline{C_{1,t,u,v+λu}^{(4)}_{λ∈𝔽}}; \underline{C_{s+1,t,u,v}^{(4)}_{sv+t^{2}+tu=α}}_{s,u,α∈𝔽;s≠0}; \underline{D_{s,t,u}^{(4)}_{s^{2}+st+u=α}}_{α,t∈𝔽}; \underline{B_{s,t}^{(4)}_{s+s^{2}+t=α}}_{α∈𝔽}; \underline{C_{r+1,λ}^{(3)}_{λ∈𝔽}}_{r≠0}; \underline{A_{1,t+u}^{(3)}}; \underline{A_{s+1,λ}^{(3)}_{λ∈𝔽}}_{s≠0}; \underline{B_{λ}^{(3)}_{λ∈𝔽}}; \underline{D_{λ}^{(3)}_{λ∈𝔽}}; \underline{A_{r+1}^{(2)}}_{r≠0}; \underline{A_{1}^{(2)}}; \underline{B^{(2)}}; \underline{A^{(1)}}]

A basis for 𝔽G_{1} is:

1

x̄  ȳ
x̄^{2}  x̄ȳ  ȳ^{2}
x̄^{3}  x̄^{2}ȳ  x̄ȳ^{2}  ȳ^{3}
x̄^{3}ȳ  x̄^{2}ȳ^{2}  x̄ȳ^{3}
x̄^{3}ȳ^{2}  x̄^{2}ȳ^{3}
x̄^{3}ȳ^{3}

We denote an arbitrary element of 𝔽G_{1} by specifying its coefficients as follows:

A = \begin{aligned}& & & a \\ & & b & & c \\ & d & & e & & f \\ g & & h & & i & & j \\ & k & & l & & m \\ & & n & & p \\ & & & q \end{aligned}

The left and right action of the generators x̄ and ȳ on such an element are:

x̄A = \begin{aligned}& & & · \\ & & a & & · \\ & b & & c & & · \\ d & & e & & f & & · \\ & h & & i & & j \\ & & l & & m \\ & & & p \end{aligned}

ȳA =
·
·  a
b  b  b+c
b  b+d+e  b+e  b+e+f
b+e+g  b+e+g+h+i  b+e+i
b+e+g+i+k  b+e+g+i+k+l+m

b+e+g+i+k+m+n

Ax̄ = ·

a  ·
b+c  c  c
c+d+e  c+e  c+e+f  c
c+e+h  c+e+h+i+j  c+e+j
c+e+h+j+k+l+m  c+e+h+j+m

c+e+h+j+k+m+p

Aȳ = \begin{aligned}& & & · \\ & & · & & a \\ & · & & b & & c \\ · & & d & & e & & f \\ & g & & h & & i \\ & & k & & l \\ & & & n \end{aligned}

The uniserial module of length 1:

A^{(1)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & · & & · \\ & & & 1 \end{aligned} ,  x̄A^{(1)} = ȳA^{(1)} = A^{(1)}x̄ = A^{(1)}ȳ = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & · & & · \\ & & & · \end{aligned}

∥G_{1}∥ = A^{(1)}

The uniserial modules of length 2:

A_{r}^{(2)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & r & & 1 \\ & & & · \end{aligned} ,  x̄A_{r}^{(2)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & · & & · \\ & & & 1 \end{aligned} ,  ȳA_{r}^{(2)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & · & & · \\ & & & r \end{aligned}

𝔽G_{1}∥⟨x^{2}, y^{2}, xy⟩∥ = 𝔽G_{1}A_{1}^{(2)}, ∥⟨y, x^{2}⟩∥ = A_{0}^{(2)}

B^{(2)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & 1 & & · \\ & & & · \end{aligned} ,  x̄B^{(2)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & · & & · \\ & & & · \end{aligned} ,  ȳB^{(2)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & · & & · \\ & & & 1 \end{aligned}

𝔽G_{1}∥⟨x, y^{2}⟩∥ = 𝔽G_{1}B^{(2)}

The uniserial modules of length 3:

A_{r,s}^{(3)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & r & & r & & 1 \\ & & s & & · \\ & & & · \end{aligned} ,  x̄A_{r,s}^{(3)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & r & & 1 \\ & & & · \end{aligned} ,  ȳA_{r,s}^{(3)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & r & & 1 \\ & & & r + 1 + s \end{aligned}

A_{r,s}^{(3)}x̄ = \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & 1 & & 1 \\ & & & r + 1 \end{aligned} , A_{r,s}^{(3)}ȳ = \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & r & & r \\ & & & s \end{aligned}

Homomorphic images: 𝔽G_{1}A_{r,s}^{3}/𝔽G_{1}A^{1} ≅ 𝔽G_{1}A_{1}^{2}. Isomorphisms: If r ≠ 1 then 𝔽G_{1}A_{r,s}^{3} ≅ 𝔽G_{1}A_{r,0}^{3}, ∀s ∈ 𝔽. 𝔽G_{1}A_{1,s}^{3} forms its own isomorphism class.

C_{s,t}^{(3)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & s & & 1 & & 1 \\ & & t & & · \\ & & & · \end{aligned} ,  x̄C_{s,t}^{(3)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & 1 & & 1 \\ & & & · \end{aligned} ,  ȳC_{s,t}^{(3)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & s & & s \\ & & & s + 1 + t \end{aligned}

C_{s,t}^{(3)}x̄ = \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & s & & 1 \\ & & & s + 1 \end{aligned} , C_{s,t}^{(3)}ȳ = \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & s & & 1 \\ & & & t \end{aligned}

Note: C_{1,s}^{(3)} = A_{1,s}^{(3)}. Homomorphic images: 𝔽G_{1}C_{s,t}^{3}/𝔽G_{1}A^{1} ≅ 𝔽G_{1}A_{s}^{2}.

Isomorphisms: If s ≠ 1 then 𝔽G_{1}C_{s,t}^{3} ≅ 𝔽G_{1}C_{s,0}^{3}. 𝔽G_{1}C_{1,t}^{3} has been mentioned above.

D_{s}^{(3)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & 1 & & · & & · \\ & & · & & s \\ & & & · \end{aligned} ,  x̄D_{s}^{(3)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & · & & · \\ & & & s \end{aligned} ,  ȳD_{s}^{(3)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & 1 & & 1 \\ & & & 1 \end{aligned}

D_{s}^{(3)}x̄ = \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & 1 & & · \\ & & & 1 + s \end{aligned} , D_{s}^{(3)}ȳ = \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & 1 & & · \\ & & & · \end{aligned}

Homomorphic images: 𝔽G_{1}D_{s}^{3}/𝔽G_{1}A^{1} ≅ 𝔽G_{1}B^{2}. Isomorphisms: 𝔽G_{1}D_{s}^{3} ≅ 𝔽G_{1}D_{0}^{3}

B_{s}^{(3)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & 1 & & 1 & & · \\ & & · & & s \\ & & & · \end{aligned} ,  x̄B_{s}^{(3)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & 1 & & · \\ & & & s \end{aligned} ,  ȳB_{s}^{(3)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & 1 & & · \\ & & & 1 \end{aligned}

B_{s}^{(3)}x̄ = \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & · & & · \\ & & & 1 + s \end{aligned} , B_{s}^{(3)}ȳ = \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & 1 & & 1 \\ & & & · \end{aligned}

Homomorphic images: 𝔽G_{1}B_{s}^{3}/𝔽G_{1}A^{1} ≅ 𝔽G_{1}A_{1}^{2}.

Isomorphisms: 𝔽G_{1}B_{s}^{3} ≅ 𝔽G_{1}B_{0}^{3} The uniserial modules of length 4:

A_{r,s,t}^{(4)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ r^{2} & & r & & r & & 1 \\ & r^{2}+r & & s & & · \\ & & t & & · \\ & & & · \end{aligned} ,  x̄A_{r,s,t}^{(4)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & r & & r & & 1 \\ & & s & & · \\ & & & · \end{aligned} ,  ȳA_{r,s,t}^{(4)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & r^{2} & & r^{2} & & r \\ & & · & & s \\ & & & t \end{aligned}

A_{r,s,t}^{(4)}x̄ = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & r & & 1 & & 1 \\ & & 1 + r^{2} + s & & r + 1 \\ & & & 1 + r^{2} \end{aligned} ,  A_{r,s,t}^{(4)}ȳ = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & r^{2} & & r & & r \\ & & r^{2} + r & & s \\ & & & t \end{aligned}

Homomorphic images: 𝔽G_{1}A_{r,s,t}^{(4)}/𝔽G_{1}A^{1} ≅ 𝔽G_{1}C_{r,r^{2}+r+s}^{(3)}. Isomorphisms: If r ≠ 1, then 𝔽G_{1}A_{r,s,t}^{(4)} ≅ 𝔽G_{1}A_{r,0,s+t+s^{2}/(r+1)}^{(4)}. 𝔽G_{1}A_{1,s,t}^{(4)} forms its own isomorphism class. 𝔽G_{1}∥⟨x^{−1}y, x^{2}y^{2}⟩∥ = 𝔽G_{1}A_{1,0,0}^{(4)} = 𝔽G_{1}C_{1,0,0,0}^{(4)} 𝔽G_{1}∥⟨xy, x^{2}y^{2}⟩∥ = 𝔽G_{1}A_{1,1,1}^{(4)} = 𝔽G_{1}C_{1,1,0,1}^{(4)} ∥⟨y⟩∥ = A_{0,0,0}^{(4)}, 𝔽G_{1}∥⟨x^{2}y⟩∥ = 𝔽G_{1}A_{0,1,0}^{(4)}

C_{s,t,u,v}^{(4)} = \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ s & & s & & 1 & & 1 \\ & u & & t & & · \\ & & v & & · \\ & & & · \end{aligned} ,  x̄C_{s,t,u,v}^{(4)} = \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & s & & 1 & & 1 \\ & & t & & · \\ & & & · \end{aligned} ,

ȳC_{s,t,u,v}^{(4)} = \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & s & & 1 & & 1 \\ & & s + 1 + u & & s + 1 + u + t \\ & & & s + 1 + u + v \end{aligned}

C_{s,t,u,v}^{(4)}x̄ = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & s & & s & & 1 \\ & & s + 1 + u + t & & s + 1 \\ & & & s + 1 + u \end{aligned} ,  C_{s,t,u,v}^{(4)}ȳ = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & s & & s & & 1 \\ & & u & & t \\ & & & v \end{aligned}

Note: C_{1,t,0,v}^{(4)} = A_{1,t,v}^{(4)}

Homomorphic images: 𝔽G_{1}C_{s,t,u,v}^{(4)}/𝔽G_{1}A^{1} ≅ 𝔽G_{1}A_{s,1+u+s^{2}+t}^{(3)} ≅ 𝔽G_{1}A_{s,t+u}^{(3)}.

Isomorphisms: For any λ ∈ 𝔽, 𝔽G_{1}C_{s+1,t,u,v}^{(4)} ≅ 𝔽G_{1}C_{s+1,t+λs,u,v+λu+λ^{2}s}^{(4)}. If s = 0,

this simplifies to 𝔽G_{1}C_{s+1,t,u,v}^{(4)} = 𝔽G_{1}C_{1,t,u,v}^{(4)} ≅ 𝔽G_{1}C_{1,t,u,v+λu}^{(4)} for all λ ∈ 𝔽.

𝔽G_{1}∥⟨x^{−1}y, y^{2}⟩∥ = 𝔽G_{1}C_{0,0,0,0}^{(4)}, 𝔽G_{1}∥⟨xy, y^{2}⟩∥ = 𝔽G_{1}C_{0,1,0,1}^{(4)}

D_{s,t,u}^{(4)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ 1 & & 1 & & · & & · \\ & · & & s+t & & t \\ & & · & & u \\ & & & · \end{aligned}  x̄D_{s,t,u}^{(4)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & 1 & & · & & · \\ & & s+t & & t \\ & & & u \end{aligned}  ȳD_{s,t,u}^{(4)} = ; \begin{aligned}& & & · \\ & & · & · \\ & · & & · & · \\ · & & · & · & & · \\ & 1 & & · & · \\ & & 1 & 1+s \\ & & & 1+t \end{aligned}

D_{s,t,u}^{(4)}x̄ = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & 1 & & 1 & & · \\ & & 1 + s & & 1 + t \\ & & & 1 + t + u \end{aligned} ,  D_{s,t,u}^{(4)}ȳ = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & 1 & & 1 & & · \\ & & · & & s + t \\ & & & · \end{aligned}

Homomorphic images: 𝔽G_{1}D_{s,t,u}^{(4)}/𝔽G_{1}A^{1} ≅ 𝔽G_{1}B_{s+t}^{(3)} ≅ 𝔽G_{1}B_{0}^{(3)}.

Isomorphisms: For all λ ∈ 𝔽, 𝔽G_{1}D_{s,t,u}^{(4)} ≅ 𝔽G_{1}D_{λ+s,t,u+λt+λ^{2}}^{(4)}.

𝔽G_{1}∥⟨xy, x^{2}⟩∥ = 𝔽G_{1}D_{0,0,0}^{(4)}, 𝔽G_{1}∥⟨xy^{−1}, x^{2}⟩∥ = 𝔽G_{1}D_{1,0,1}^{(4)}

B_{s,t}^{(4)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ 1 & & · & & · & & · \\ & · & & 1+s & & · \\ & & · & & t \\ & & & · \end{aligned} ,  x̄B_{s,t}^{(4)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & 1+s & & · \\ & & & t \end{aligned} ,  ȳB_{s,t}^{(4)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & 1 & & 1 & & · \\ & & 1 & & s \\ & & & 1 \end{aligned}
B_{s,t}^{(4)}x̄ = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & 1 + s & & · \\ & & & t \end{aligned} ,  B_{s,t}^{(4)}ȳ = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & 1 & & · & & · \\ & & · & & 1 + s \\ & & & · \end{aligned}

Homomorphic images: 𝔽G_{1}B_{s,t}^{(4)}/𝔽G_{1}A^{1} ≅ 𝔽G_{1}D_{1+s}^{(3)} ≅ 𝔽G_{1}D_{0}^{(3)}. Isomorphisms: for any λ ∈ 𝔽, 𝔽G_{1}B_{s,t}^{(4)} ≅ 𝔽G_{1}B_{s+λ,t+λ+λ^{2}}^{(4)} ≅ 𝔽G_{1}B_{0,s+s^{2}+t}^{(4)}. ∥⟨x⟩∥ = B_{1,0}^{(4)}, 𝔽G_{1}∥⟨xy^{2}⟩∥ = 𝔽G_{1}B_{0,0}^{(4)}

There are no uniserial modules of length 5.

§2 Uniserial submodules of 𝔽(2 × 4).2 Group: (2 × 4).2 = ⟨x, y | x^{4} = y^{4} = 1, ^{x}y = y^{−1}⟩ This is the partially ordered set of uniserial 𝔽(2 × 4).2 submodules:

[figure: A_{r,s,t}^{(4)}; B_{s,t}^{(4)}; A_{r,s}^{(3)}; B_{s}^{(3)}; A_{r}^{(2)}; B^{(2)}; A^{(1)}]

This is the homomorphism diagram for uniserial 𝔽(2 × 4).2 modules:

[figure: \underline{A_{r,s+λ,t+λ(1+r+λ)}^{(4)}_{λ∈𝔽}}; \underline{B_{s,t}^{(4)}}; \underline{A_{r+1,λ}^{(3)}_{λ∈𝔽}}; \underline{B_{s}^{(3)}}; \underline{A_{r}^{(2)}}; \underline{B^{(2)}}; \underline{A^{(1)}}]

A basis for 𝔽(2 × 4).2 is:

1

x̄  ȳ
x̄^{2}  x̄ȳ  ȳ^{2}
x̄^{3}  x̄^{2}ȳ  x̄ȳ^{2}  ȳ^{3}
x̄^{3}ȳ  x̄^{2}ȳ^{2}  x̄ȳ^{3}
x̄^{3}ȳ^{2}  x̄^{2}ȳ^{3}
x̄^{3}ȳ^{3}

A = \begin{aligned}& & & a \\ & & b & & c \\ & d & & e & & f \\ g & & h & & i & & j \\ & k & & l & & m \\ & & n & & p \\ & & & q \end{aligned} ,  x̄A = \begin{aligned}& & & · \\ & & a & & · \\ & b & & c & & · \\ d & & e & & f & & · \\ & h & & i & & j \\ & & l & & m \\ & & & p \end{aligned} ,

ȳA = \begin{aligned}& & & · \\ & & · & & a \\ & · & & b & & b + c \\ · & & d & & b + e & & b + e + f \\ & g & & g + h & & b + e + i \\ & & g + k & & g + k + l \\ & & & g + k + n \end{aligned}

Ax̄ = \begin{aligned}& & & · \\ & & a & & · \\ & b & & c & & c \\ d & & e & & c + e + f & & c \\ & h & & e + h + i & & c + e + j \\ & & h + k + l & & e + h + m \\ & & & h + k + p \end{aligned}

Aȳ = \begin{aligned}& & & · \\ & & · & & a \\ & · & & b & & c \\ · & & d & & e & & f \\ & g & & h & & i \\ & & k & & l \\ & & & n \end{aligned}

The uniserial module of length 1:

A^{(1)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & · & & · \\ & & & 1 \end{aligned} ,  x̄A^{(1)} = ȳA^{(1)} = A^{(1)}x̄ = A^{(1)}ȳ = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & · & & · \\ & & & · \end{aligned}

∥(2x4).2∥ = A^{(1)}

The uniserial modules of length 2:

A_{r}^{(2)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & 1 & & r \\ & & & · \end{aligned} ,  x̄A_{r}^{(2)} = A_{r}^{(2)}x̄ = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & · & & · \\ & & & r \end{aligned} ,  ȳA_{r}^{(2)} = A_{r}^{(2)}ȳ = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & · & & · \\ & & & 1 \end{aligned}

∥⟨x, y^{2}⟩∥ = A_{0}^{(2)}, ∥⟨xy, y^{2}⟩∥ ∈ 𝔽(2 × 4).2A_{1}^{(2)}

B^{(2)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & · & & 1 \\ & & & · \end{aligned} ,  x̄B^{(2)} = B^{(2)}x̄ = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & · & & · \\ & & & 1 \end{aligned} ,  ȳB^{(2)} = B^{(2)}ȳ = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & · & & · \\ & & & · \end{aligned}

∥⟨y, x^{2}⟩∥ = B^{(2)}

The uniserial modules of length 3:

A_{r,s}^{(3)} = \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & 1 & & r + 1 & & r^{2} + r \\ & & · & & s \\ & & & · \end{aligned} ,

x̄A_{r,s}^{(3)} = \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & r + 1 & & r^{2} + r \\ & & & s \end{aligned} ,  ȳA_{r,s}^{(3)} = \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & 1 & & r \\ & & & 1 \end{aligned}

A_{r,s}^{(3)}x̄ = \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & r & & r^{2} + r \\ & & & s + 1 \end{aligned} , A_{r,s}^{(3)}ȳ = \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & 1 & & r + 1 \\ & & & · \end{aligned}

B_{s}^{(3)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & 1 \\ & & s & & · \\ & & & · \end{aligned} ,  x̄B_{s}^{(3)} = B_{s}^{(3)}x̄ = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & · & & 1 \\ & & & · \end{aligned} ,  ȳB_{s}^{(3)} = B_{s}^{(3)}ȳ = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & · & & · \\ & & & s \end{aligned}

The uniserial modules of length 4:

A_{r,s,t}^{(4)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ 1 & & r & & r^{2} + r & & r^{3} + r^{2} \\ & · & & 1 + r + s & & r^{2} + r \\ & & · & & t \\ & & & · \end{aligned} ,

x̄A_{r,s,t}^{(4)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & r & & r^{2}+r & & r^{3}+r^{2} \\ & & 1+r+s & & r^{2}+r \\ & & & t \end{aligned} ,  ȳA_{r,s,t}^{(4)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & 1 & & 1+r & & r^{2}+r \\ & & 1 & & r+s \\ & & & · \end{aligned}

A_{r,s,t}^{(4)}x̄ = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & r & & r^{2} & & r^{3} + r^{2} \\ & & s + 1 & & r^{2} \\ & & & r + t \end{aligned} ,  A_{r,s,t}^{(4)}ȳ = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & 1 & & r & & r^{2} + r \\ & & · & & r + s + 1 \\ & & & · \end{aligned}

∥⟨x⟩∥ ∈ 𝔽(2 × 4).2A_{0,1,0}^{(4)} ≅ 𝔽(2 × 4).2A_{0,0,0}^{(4)} ∋ ∥⟨xy^{2}⟩∥ ∥⟨xy⟩∥ ∈ 𝔽(2 × 4).2A_{1,0,0}^{(4)} ≅ 𝔽(2 × 4).2A_{1,1,1}^{(4)} ∋ ∥⟨xy^{−1}⟩∥

B_{s,t}^{(4)} = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & 1 \\ & · & & s & & · \\ & & t & & · \\ & & & · \end{aligned} , x̄B_{s,t}^{(4)} = B_{s,t}^{(4)}x̄ = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & 1 \\ & & s & & · \\ & & & · \end{aligned} , ȳB_{s,t}^{(4)} = B_{s,t}^{(4)}ȳ = ; \begin{aligned}& & & · \\ & & · & & · \\ & · & & · & & · \\ · & & · & & · & & · \\ & · & & · & & · \\ & & · & & s \\ & & & t \end{aligned}

∥⟨y⟩∥ = B_{0,0}^{(4)}, ∥⟨x^{2}y⟩∥ = B_{1,0}^{(4)} There are no uniserial modules of length 5.

§3 Uniserial submodules of 𝔽Mod_{16} Group: Mod_{16} = ⟨x, y | x^{8} = y^{2} = 1, ^{y}x = x^{5}⟩ This is the partially ordered set of uniserial 𝔽Mod_{16} submodules:

[figure: B_{u,v,w,x}^{(8)}; B_{0,u,v,w}^{(7)}; B_{δ,u,v,w}^{(7)}; B_{t,u,v}^{(6)}; B_{δ,u,v}^{(6)}; B_{t,u}^{(5)}; B_{0,t}^{(4)}; B_{s,t}^{(4)}; B_{s}^{(3)}; A_{r}^{(2)}; B^{(2)}; A^{(1)}]  where δ = 0 or 1

This is the homomorphism diagram for uniserial 𝔽Mod_{16} modules:

[figure: \underline{B_{u+λ,v+λ+λ^{2},w+ν+λ^{2},x+ν+λ^{4}}^{(8)}_{λ,ν∈𝔽}}; \underline{B_{1,u+1+λ,v+λ+λ^{2},ν}^{(7)}_{λ,ν∈𝔽}}; \underline{B_{δ,u+1+λ,v+λ+λ^{2},ν}^{(7)}_{λ,ν∈𝔽}}_{δ∈\{0,1\}}; \underline{B_{t,u+λ,v+λ+λ^{2}}^{(6)}_{λ∈𝔽}}; \underline{B_{δ+1,u+λ,v+λ+λ^{2}}^{(6)}_{λ∈𝔽}}_{δ∈\{0,1\}}; \underline{B_{t+1,μ}^{(5)}}_{μ∈𝔽}; \underline{B_{0,t}^{(4)}}; \underline{B_{s,t}^{(4)}}; \underline{B_{s}^{(3)}}; \underline{A_{r}^{(2)}}; \underline{B^{(2)}}; \underline{A^{(1)}}]

A basis for 𝔽Mod_{16} is

1

x̄  ȳ
x̄^{2}  x̄ȳ
x̄^{3}  x̄^{2}ȳ
x̄^{4}  x̄^{3}ȳ
x̄^{5}  x̄^{4}ȳ
x̄^{6}  x̄^{5}ȳ
x̄^{7}  x̄^{6}ȳ
x̄^{7}ȳ

A = \begin{aligned}& a \\ b & & c \\ d & & e \\ f & & g \\ h & & i \\ j & & k \\ l & & m \\ n & & p \\ & q \end{aligned} ,  x̄A = \begin{aligned}& · \\ a & & · \\ b & & c \\ d & & e \\ f & & g \\ h & & i \\ j & & k \\ l & & m \\ & p \end{aligned} ,  ȳA = \begin{aligned}& · \\ · & a \\ · & b \\ · & d \\ b & f \\ b & b+e+h \\ f & b+e+j \\ f & f +i+l \\ & f +i+n \end{aligned}

Ax̄ = \begin{aligned}& · \\ a & & · \\ b & & c \\ d & & e \\ c+f & & g \\ c+e+h & & c+i \\ e + g + j & & c + e + k \\ g + i + l & & e + g + m \\ & g + i + p \end{aligned} , Aȳ = \begin{aligned}& · \\ · & & a \\ · & & b \\ · & & d \\ · & & f \\ · & & h \\ · & & j \\ · & & l \\ & n \end{aligned}

The uniserial module of length 1:

A^{(1)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & 1 \end{aligned} ,  x̄A^{(1)} = ȳA^{(1)} = A^{(1)}x̄ = A^{(1)}ȳ = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & · \end{aligned}

A_{r}^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & r \\ & · \end{aligned} ,  x̄A_{r}^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & r \end{aligned} ,  ȳA_{r}^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & 1 \end{aligned}

A_{r}^{(2)}x̄ = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & r \end{aligned} , A_{r}^{(2)}ȳ = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & 1 \end{aligned}

B^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ & · \end{aligned} ,  x̄B^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & 1 \end{aligned} ,  ȳB^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & · \end{aligned}

B^{(2)}x̄ = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & 1 \end{aligned} , B^{(2)}ȳ = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & · \end{aligned}

B_{s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ & · \end{aligned} ,  x̄B_{s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ & · \end{aligned} ,  ȳB_{s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & s \end{aligned}

B_{s}^{(3)}x̄ = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ & · \end{aligned} , B_{s}^{(3)}ȳ = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & s \end{aligned}

The uniserial modules of length 4:

B_{s,t}^{(4)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ t & & · \\ & · \end{aligned} ,  x̄B_{s,t}^{(4)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ & · \end{aligned} ,  ȳB_{s,t}^{(4)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & s \\ & t \end{aligned}

B_{s,t}^{(4)}x̄ = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ & · \end{aligned} , B_{s,t}^{(4)}ȳ = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & s \\ & t \end{aligned}

B_{t,u}^{(5)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ · & & · \\ t & & · \\ u & & · \\ & · \end{aligned} ,  x̄B_{t,u}^{(5)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ · & & · \\ t & & · \\ & · \end{aligned} ,  ȳB_{t,u}^{(5)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 + t \\ & 1 + u \end{aligned}

B_{t,u}^{(5)}x̄ = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ · & & · \\ t + 1 & & · \\ & 1 \end{aligned} , B_{t,u}^{(5)}ȳ = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & t \\ & u \end{aligned}

The uniserial modules of length 6:

B_{t,u,v}^{(6)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & 1 \\ · & & · \\ t & & · \\ u & & · \\ v & & · \\ & · \end{aligned} ,  x̄B_{t,u,v}^{(6)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ · & & · \\ t & & · \\ u & & · \\ & · \end{aligned} ,  ȳB_{t,u,v}^{(6)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & t \\ · & & u \\ & v \end{aligned}

B_{t,u,v}^{(6)}x̄ = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ · & & · \\ t + 1 & & · \\ u + 1 & & 1 \\ & 1 \end{aligned} , B_{t,u,v}^{(6)}ȳ = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & t \\ · & & u \\ & v \end{aligned}

The uniserial modules of length 7:

Note: δ = 0 or 1.

B_{δ,u,v,w}^{(7)} = \begin{aligned}& · \\ · & & · \\ · & & 1 \\ · & & · \\ δ & & · \\ u & & · \\ v & & · \\ w & & · \\ & · \end{aligned} ,  x̄B_{δ,u,v,w}^{(7)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & 1 \\ · & & · \\ δ & & · \\ u & & · \\ v & & · \\ & · \end{aligned} ,  ȳB_{δ,u,v,w}^{(7)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 + δ \\ · & & 1 + u \\ · & & v \\ & w \end{aligned}

B_{δ,u,v,w}^{(7)}x̄ = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & 1 \\ · & & · \\ δ + 1 & & · \\ u + 1 & & 1 \\ v & & 1 \\ & · \end{aligned} , B_{δ,u,v,w}^{(7)}ȳ = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & δ \\ · & & u \\ · & & v \\ & w \end{aligned}

The uniserial modules of length 8:

B_{u,v,w,x}^{(8)} = \begin{aligned}& · \\ · & & 1 \\ · & & · \\ · & & · \\ u & & · \\ v & & · \\ w & & · \\ x & & · \\ & · \end{aligned} ,  x̄B_{u,v,w,x}^{(8)} = \begin{aligned}& · \\ · & & · \\ · & & 1 \\ · & & · \\ · & & · \\ u & & · \\ v & & · \\ w & & · \\ & · \end{aligned} ,  ȳB_{u,v,w,x}^{(8)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & u \\ · & & v \\ · & & w \\ & x \end{aligned}

B_{u,v,w,x}^{(8)}x̄ = \begin{aligned}& · \\ · & & · \\ · & & 1 \\ · & & · \\ 1 & & · \\ u + 1 & & 1 \\ v & & 1 \\ w & & · \\ & · \end{aligned} , B_{u,v,w,x}^{(8)}ȳ = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & u \\ · & & v \\ · & & w \\ & x \end{aligned}

There are no uniserial modules of length 9.

§4 Uniserial submodules of 𝔽D_{16} Group: D_{16} = ⟨x, y | x^{2} = y^{2} = (xy)^{8} = 1⟩ This is the partially ordered set of uniserial 𝔽D_{16} submodules:

[figure: A_{s,u,w,x}^{(8)}; B_{s,u,w,x}^{(8)}; A_{s,u,w}^{(7)}; B_{s,u,w}^{(7)}; A_{s,u,0}^{(6)}; A_{s,u,v}^{(6)}; B_{s,u,v}^{(6)}; B_{s,u,0}^{(6)}; A_{s,u}^{(5)}; B_{s,u}^{(5)}; A_{s,t}^{(4)}; A_{s,0}^{(4)}; B_{s,0}^{(4)}; B_{s,t}^{(4)}; A_{s}^{(3)}; B_{s}^{(3)}; A_{0}^{(2)}; A_{r}^{(2)}; B^{(2)}; A^{(1)}]

This is the homomorphism diagram for uniserial 𝔽D_{16} modules:

[figure: \underline{B_{λ,μ,ν,x}^{(8)}_{λ,μ,ν∈𝔽}}; \underline{A_{λ,μ,ν,x}^{(8)}_{λ,μ,ν∈𝔽}}; \underline{A_{λ,μ,ν}^{(7)}_{λ,μ,ν∈𝔽}}; \underline{B_{λ,μ,ν}^{(7)}_{λ,μ,ν∈𝔽}}; \underline{B_{λ,μ,0}^{(6)}_{λ,μ∈𝔽}}; \underline{B_{λ,μ,v}^{(6)}_{λ,μ∈𝔽}}; \underline{A_{λ,μ,v}^{(6)}_{λ,μ∈𝔽}}; \underline{A_{λ,μ,0}^{(6)}_{λ,μ∈𝔽}}; \underline{A_{λ,μ}^{(5)}_{λ,μ∈𝔽}}; \underline{B_{λ,μ}^{(5)}_{λ,μ∈𝔽}}; \underline{B_{λ,0}^{(4)}_{λ∈𝔽}}; \underline{B_{λ,t}^{(4)}_{λ∈𝔽}}; \underline{A_{λ,t}^{(4)}_{λ∈𝔽}}; \underline{A_{λ,0}^{(4)}_{λ∈𝔽}}; \underline{A_{λ}^{(3)}_{λ∈𝔽}}; \underline{B_{λ}^{(3)}_{λ∈𝔽}}; \underline{B^{(2)}}; \underline{A_{r}^{(2)}}; \underline{A_{0}^{(2)}}; \underline{A^{(1)}}]

For this group ring we use a basis which is not in Jennings’ form. The group G = D_{2n} = ⟨x, y | x^{2} = y^{2} = (xy)^{n} = 1⟩ is the dihedral group of order 2n. We write x̄ = x−1 and ȳ = y −1. Let n be a power of 2, and 𝔽 a field of characteristic 2. Let a_{m} = x̄ȳx̄ ⋯ be an element of 𝔽D_{2n}, where x̄ and ȳ appear alternately in the product, starting with x̄, and the total number of terms in this product is m. Let b_{m} = ȳx̄ȳ⋯ be defined in a like manner, where the product starts with ȳ. We define a_{0} = b_{0} = 1. It may be shown that

i) a_{n} = b_{n} = ∥D_{2n}∥ ii) a_{m} = b_{m} = 0 when m > n iii) a_{m} and b_{m} are elements of Rad^{m}(𝔽D_{2n} iv) a_{m} and b_{m} are not elements of Rad^{m+1}(𝔽D_{2n} if m ≤ n

v) The set {a_{0}, a_{i}, b_{i}, a_{n} | 0 < i < n} forms a basis for 𝔽D_{2n} filtered with respect to the powers of the radical, where the subscript indicates the appropriate power of the radical. We choose not to prove these statements here. The following results may be quickly generalized to other dihedral 2-groups. A basis for 𝔽D_{16} is

1

x̄  ȳ
ȳx̄  x̄ȳ
x̄ȳx̄  ȳx̄ȳ
ȳx̄ȳx̄  x̄ȳx̄ȳ
x̄ȳx̄ȳx̄  ȳx̄ȳx̄ȳ
ȳx̄ȳx̄ȳx̄  x̄ȳx̄ȳx̄ȳ
x̄ȳx̄ȳx̄ȳx̄  ȳx̄ȳx̄ȳx̄ȳ
ȳx̄ȳx̄ȳx̄ȳ

A = \begin{aligned}& a \\ b & & c \\ d & & e \\ f & & g \\ h & & i \\ j & & k \\ l & & m \\ n & & p \\ & q \end{aligned}  x̄A = \begin{aligned}& · \\ a & & · \\ · & & c \\ d & & · \\ · & & g \\ h & & · \\ · & & k \\ l & & · \\ & p \end{aligned}  ȳA = \begin{aligned}& · \\ · & & a \\ b & & · \\ · & & e \\ f & & · \\ · & & i \\ j & & · \\ · & & m \\ & n \end{aligned}

Ax̄ = \begin{aligned}& · \\ a & & · \\ c & & · \\ e & & · \\ g & & · \\ i & & · \\ k & & · \\ m & & · \\ & p \end{aligned} Aȳ = \begin{aligned}& · \\ · & & a \\ · & & b \\ · & & d \\ · & & f \\ · & & h \\ · & & j \\ · & & l \\ & n \end{aligned}

The Uniserial module of length 1:

A^{(1)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & 1 \end{aligned}  x̄A^{(1)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & · \end{aligned}  ȳA^{(1)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & · \end{aligned}

The Uniserial modules of length 2:

A_{r}^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & r \\ & · \end{aligned}  x̄A_{r}^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & r \end{aligned}  ȳA_{r}^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & 1 \end{aligned}

B^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ & · \end{aligned}  x̄B^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & 1 \end{aligned}  ȳB^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & · \end{aligned}

A_{s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & · \\ · & & s \\ & · \end{aligned}  x̄A_{s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & · \\ & s \end{aligned}  ȳA_{s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & · \end{aligned}

B_{s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ & · \end{aligned}  x̄B_{s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & · \end{aligned}  ȳB_{s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ & s \end{aligned}

The Uniserial modules of length 4:

A_{s,t}^{(4)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & · \\ · & & s \\ · & & t \\ & · \end{aligned}  x̄A_{s,t}^{(4)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & t \end{aligned}  ȳA_{s,t}^{(4)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & · \\ · & & s \\ & · \end{aligned}

B_{s,t}^{(4)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ t & & · \\ & · \end{aligned}  x̄B_{s,t}^{(4)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ & · \end{aligned}  ȳB_{s,t}^{(4)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & t \end{aligned}

A_{s,u}^{(5)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & · \\ · & & s \\ · & & · \\ · & & u \\ & · \end{aligned}  x̄A_{s,u}^{(5)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & · \\ · & & s \\ · & & · \\ & u \end{aligned}  ȳA_{s,u}^{(5)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & · \end{aligned}

B_{s,u}^{(5)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ · & & · \\ u & & · \\ & · \end{aligned}  x̄B_{s,u}^{(5)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & · \end{aligned}  ȳB_{s,u}^{(5)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ · & & · \\ & u \end{aligned}

The Uniserial modules of length 6:

A_{s,u,v}^{(6)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ 1 & & · \\ · & & s \\ · & & · \\ · & & u \\ · & & v \\ & · \end{aligned}  x̄A_{s,u,v}^{(6)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & v \end{aligned}  ȳA_{s,u,v}^{(6)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & · \\ · & & s \\ · & & · \\ · & & u \\ & · \end{aligned}

B_{s,u,v}^{(6)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ · & & · \\ u & & · \\ v & & · \\ & · \end{aligned}  x̄B_{s,u,v}^{(6)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ · & & · \\ u & & · \\ & · \end{aligned}  ȳB_{s,u,v}^{(6)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & v \end{aligned}

A_{s,u,w}^{(7)} = \begin{aligned}& · \\ · & & · \\ 1 & & · \\ · & & s \\ · & & · \\ · & & u \\ · & & · \\ · & & w \\ & · \end{aligned}  x̄A_{s,u,w}^{(7)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ 1 & & · \\ · & & s \\ · & & · \\ · & & u \\ · & & · \\ & w \end{aligned}  ȳA_{s,u,w}^{(7)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & · \end{aligned}

B_{s,u,w}^{(7)} = \begin{aligned}& · \\ · & & · \\ · & & 1 \\ s & & · \\ · & & · \\ u & & · \\ · & & · \\ w & & · \\ & · \end{aligned}  x̄B_{s,u,w}^{(7)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & · \end{aligned}  ȳB_{s,u,w}^{(7)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ · & & · \\ u & & · \\ · & & · \\ & w \end{aligned}

The Uniserial modules of length 8:

A_{s,u,w,x}^{(8)} = \begin{aligned}& · \\ 1 & & · \\ · & & s \\ · & & · \\ · & & u \\ · & & · \\ · & & w \\ · & & x \\ & · \end{aligned}  x̄A_{s,u,w,x}^{(8)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & x \end{aligned}  ȳA_{s,u,w,x}^{(8)} = \begin{aligned}& · \\ · & & · \\ 1 & & · \\ · & & s \\ · & & · \\ · & & u \\ · & & · \\ · & & w \\ & · \end{aligned}

B_{s,u,w,x}^{(8)} = \begin{aligned}& · \\ · & & 1 \\ s & & · \\ · & & · \\ u & & · \\ · & & · \\ w & & · \\ x & & · \\ & · \end{aligned}  x̄B_{s,u,w,x}^{(8)} = \begin{aligned}& · \\ · & & · \\ · & & 1 \\ s & & · \\ · & & · \\ u & & · \\ · & & · \\ w & & · \\ & · \end{aligned}  ȳB_{s,u,w,x}^{(8)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & x \end{aligned}

There is no uniserial module of length 9.

§5 Uniserial submodules of 𝔽SD_{16} Group: SD_{16} = ⟨x, y | x^{8} = y^{2} = 1, ^{y}x = x^{3}⟩ This is the partially ordered set of uniserial 𝔽SD_{16} submodules:

[figure: B_{s,u,w,x}^{(8)}; B_{s,u,w}^{(7)}; A_{s,u,w}^{(7)}; B_{s,u,s^{4}+s^{3}}^{(6)}; B_{s,u,v}^{(6)}; A_{s,u,v}^{(6)}; A_{s,u,u}^{(6)}; B_{s,u}^{(5)}; A_{s,u}^{(5)}; B_{s,t}^{(4)}; B_{s,s^{2}+s}^{(4)}; A_{s,1}^{(4)}; A_{s,t}^{(4)}; B_{s}^{(3)}; A_{s}^{(3)}; B^{(2)}; A_{r}^{(2)}; A_{1}^{(2)}; A^{(1)}]

This is the homomorphism diagram for uniserial 𝔽SD_{16} modules:

[figure: \underline{B_{s,u,w,x}^{(8)}_{s^{2}+s^{3}+s^{4}+s^{5}+s^{2}u+u+u^{2}+w+x=α}}_{α∈𝔽}; \underline{A_{λ,μ,ν}^{(7)}_{λ,μ,ν∈𝔽}}; \underline{B_{λ,μ,ν}^{(7)}_{λ,μ,ν∈𝔽}}; \underline{B_{λ,μ,λ^{3}+λ^{4}}^{(6)}_{λ,μ∈𝔽}}; \underline{B_{s+λ,μ,v+sλ^{2}+s^{2}λ+λ^{3}+λ^{4}}^{(6)}_{λ,μ∈𝔽}}; \underline{A_{λ,u+μ,v+μ}^{(6)}_{λ,μ∈𝔽}}; \underline{A_{λ,μ,μ}^{(6)}_{λ,μ∈𝔽}}; \underline{A_{λ,μ}^{(5)}_{λ,μ∈𝔽}}; \underline{B_{λ,μ}^{(5)}_{λ,μ∈𝔽}}; \underline{B_{λ,λ+λ^{2}}^{(4)}_{λ∈𝔽}}; \underline{B_{s+λ,λ+λ^{2}+t}^{(4)}_{λ∈𝔽}}; \underline{A_{λ,t}^{(4)}_{λ∈𝔽}}; \underline{A_{λ,1}^{(4)}_{λ∈𝔽}}; \underline{A_{λ}^{(3)}_{λ∈𝔽}}; \underline{B_{λ}^{(3)}_{λ∈𝔽}}; \underline{A_{r}^{(2)}}; \underline{A_{1}^{(2)}}; \underline{B^{(2)}}; \underline{A^{(1)}}]

A basis for 𝔽SD_{16} is

1

x̄  ȳ
x̄^{2}  x̄ȳ
x̄^{3}  x̄^{2}ȳ
x̄^{4}  x̄^{3}ȳ
x̄^{5}  x̄^{4}ȳ
x̄^{6}  x̄^{5}ȳ
x̄^{7}  x̄^{6}ȳ
x̄^{7}ȳ

A = \begin{aligned}& a \\ b & & c \\ d & & e \\ f & & g \\ h & & i \\ j & & k \\ l & & m \\ n & & p \\ & q \end{aligned} ,  x̄A = \begin{aligned}& · \\ a & & · \\ b & & c \\ d & & e \\ f & & g \\ h & & i \\ j & & k \\ l & & m \\ & p \end{aligned} ,

ȳA = \begin{aligned}& · \\ · & & a \\ b & & b \\ b & & b + d + e \\ d + f & & b + e + f \\ · & & d + f + g + h + i \\ d + f + j & & j \\ j & & d + f + g + i + j + l + m \\ & j + m + n \end{aligned}

The Uniserial modules of length 1:

A^{(1)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & 1 \end{aligned} ,  x̄A^{(1)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & · \end{aligned} ,  ȳA^{(1)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & · \end{aligned}

A_{r}^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & r \\ & · \end{aligned} ,  x̄A_{r}^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & r \end{aligned} ,  ȳA_{r}^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & 1 \end{aligned}

B^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ & · \end{aligned} ,  x̄B^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & 1 \end{aligned} ,  ȳB^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & · \end{aligned}

The Uniserial modules of length 3:

A_{s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & 1 \\ · & & s \\ & · \end{aligned} ,  x̄A_{s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & 1 \\ & s \end{aligned} ,  ȳA_{s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & 1 \end{aligned}

B_{s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ & · \end{aligned} ,  x̄B_{s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ & · \end{aligned} ,  ȳB_{s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ & 1 + s \end{aligned}

A_{s,t}^{(4)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & 1 \\ · & & s \\ · & & t \\ & · \end{aligned} ,  x̄A_{s,t}^{(4)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & 1 \\ · & & s \\ & t \end{aligned} ,  ȳA_{s,t}^{(4)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & 1 \\ 1 & & 1 + s \\ & 1 + s \end{aligned}

B_{s,t}^{(4)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ t & & · \\ & · \end{aligned} ,  x̄B_{s,t}^{(4)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ & · \end{aligned} ,  ȳB_{s,t}^{(4)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & s \\ & t \end{aligned}

The Uniserial modules of length 5:

A_{s,u}^{(5)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & 1 \\ · & & s \\ · & & 1 \\ · & & u \\ & · \end{aligned} ,  x̄A_{s,u}^{(5)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & 1 \\ · & & s \\ · & & 1 \\ & u \end{aligned} ,  ȳA_{s,u}^{(5)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & 1 \end{aligned}

B_{s,u}^{(5)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ s^{2} + s & & · \\ u & & · \\ & · \end{aligned} ,  x̄B_{s,u}^{(5)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ s^{2} + s & & · \\ & · \end{aligned} ,  ȳB_{s,u}^{(5)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & s \\ s & & 1 + s^{2} \\ & s + u \end{aligned}

A_{s,u,v}^{(6)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ 1 & & 1 \\ · & & s \\ · & & 1 \\ · & & u \\ · & & v \\ & · \end{aligned} ,  x̄A_{s,u,v}^{(6)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & 1 \\ · & & s \\ · & & 1 \\ · & & u \\ & v \end{aligned} ,  ȳA_{s,u,v}^{(6)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & 1 \\ · & & s \\ 1 & & · \\ · & & s + u \\ & u \end{aligned}

B_{s,u,v}^{(6)} = ; \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ s^{2} + s & & · \\ u & & · \\ v & & · \\ & · \end{aligned} ,  x̄B_{s,u,v}^{(6)} = ; \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ s^{2} + s & & · \\ u & & · \\ & · \end{aligned} ,  ȳB_{s,u,v}^{(6)} = ; \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 + s \\ s^{2} + s & & s^{2} + s \\ s^{2} + s & & 1 + s^{2} + s + u \\ & s^{2} + s + v \end{aligned}

The Uniserial modules of length 7:

A_{s,u,w}^{(7)} = \begin{aligned}& · \\ · & & · \\ 1 & & 1 \\ · & & s \\ · & & 1 \\ · & & u \\ · & & u \\ · & & w \\ & · \end{aligned} ,  x̄A_{s,u,w}^{(7)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ 1 & & 1 \\ · & & s \\ · & & 1 \\ · & & u \\ · & & u \\ & w \end{aligned} ,  ȳA_{s,u,w}^{(7)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & 1 \\ · & & s \\ 1 & & · \\ · & & s + u \\ & u \end{aligned}

B_{s,u,w}^{(7)} = ; \begin{aligned}& · \\ · & & · \\ · & & 1 \\ s & & · \\ s^{2} + s & & · \\ u & & · \\ s^{4} + s^{3} & & · \\ w & & · \\ & · \end{aligned} ,  x̄B_{s,u,w}^{(7)} = ; \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ s^{2} + s & & · \\ u & & · \\ s^{4} + s^{3} & & · \\ & · \end{aligned} ,  ȳB_{s,u,w}^{(7)} = ; \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & 1 + s \\ · & & s^{2} \\ s + u & & u \\ u & & s + u + s^{4} + s^{3} \\ & u + w \end{aligned}

The Uniserial modules of length 8:

B_{s,u,w,x}^{(8)} = \begin{aligned}& · \\ · & & 1 \\ s & & · \\ s^{2} + s & & · \\ u & & · \\ s^{4} + s^{3} & & · \\ w & & · \\ x & & · \\ & · \end{aligned} , x̄B_{s,u,w,x}^{(8)} = \begin{aligned}& · \\ · & & · \\ · & & 1 \\ s & & · \\ s^{2} + s & & · \\ u & & · \\ s^{4} + s^{3} & & · \\ w & & · \\ & · \end{aligned} ,

ȳB_{s,u,w,x}^{(8)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & s \\ s^{2} & & s^{2} + s \\ · & & s^{2} + u \\ s^{2} + s^{4} + s^{3} & & s^{4} + s^{3} \\ s^{4} + s^{3} & & s^{2} + s^{4} + s^{3} + w \\ & s^{4} + s^{3} + x \end{aligned}

There is no uniserial module of length 9.

§6 Uniserial submodules of 𝔽Q_{16} Group: Q_{16} = ⟨x, y | x^{8} = 1, x^{4} = y^{2}, ^{y}x = x^{−1}⟩ This is the partially ordered set of uniserial 𝔽Q_{16} submodules:

[figure: A_{s,u,w}^{(7)}; B_{s,u,w}^{(7)}; A_{s,u,s^{2}+s+u+1}^{(6)}; A_{s,u,v}^{(6)}; B_{s,u,v}^{(6)}; B_{s,u,s^{4}+s^{3}+s^{2}+1}^{(6)}; A_{s,t,u}^{(5)}; A_{s,1,u}^{(5)}; B_{s,u}^{(5)}; A_{s,t}^{(4)}; B_{s,s^{2}+s+1}^{(4)}; B_{s,t}^{(4)}; A_{s}^{(3)}; B_{s}^{(3)}; A_{1}^{(2)}; A_{r}^{(2)}; B^{(2)}; A^{(1)}]

This is the homomorphism diagram for uniserial 𝔽Q_{16} modules:

[figure: \underline{B_{λ,μ,ν}^{(7)}_{λ,μ,ν∈𝔽}}; \underline{A_{λ,μ,ν}^{(7)}_{λ,μ,ν∈𝔽}}; \underline{A_{s,u,v}^{(6)}_{s+s^{2}+u+v=1}}; \underline{A_{s,u,v}^{(6)}_{s+s^{2}+u+v=α}}_{α∈𝔽}; \underline{B_{s,u,v}^{(6)}_{s^{2}+s^{3}+s^{4}+v=α}}_{α∈𝔽}; \underline{B_{s,u,v}^{(6)}_{s^{2}+s^{3}+s^{4}+v=1}}; \underline{B_{λ,μ}^{(5)}_{λ,μ∈𝔽}}; \underline{A_{λ,μ}^{(5)}_{λ,μ∈𝔽}}; \underline{A_{λ,1}^{(4)}_{λ∈𝔽}}; \underline{A_{λ,t}^{(4)}_{λ∈𝔽}}; \underline{B_{s,t}^{(4)}_{s+s^{2}+t=α}}_{α∈𝔽}; \underline{B_{s,t}^{(4)}_{s+s^{2}+t=1}}; \underline{B_{λ}^{(3)}_{λ∈𝔽}}; \underline{A_{λ}^{(3)}_{λ∈𝔽}}; \underline{A_{1}^{(2)}}; \underline{A_{r}^{(2)}}; \underline{B^{(2)}}; \underline{A^{(1)}}]

A basis for 𝔽Q_{16} is

1

x̄  ȳ
x̄^{2}  x̄ȳ
x̄^{3}  x̄^{2}ȳ
x̄^{4}  x̄^{3}ȳ
x̄^{5}  x̄^{4}ȳ
x̄^{6}  x̄^{5}ȳ
x̄^{7}  x̄^{6}ȳ
x̄^{7}ȳ
A = \begin{aligned}& a \\ b & & c \\ d & & e \\ f & & g \\ h & & i \\ j & & k \\ l & & m \\ n & & p \\ & q \end{aligned} ,  x̄A = \begin{aligned}& · \\ a & & · \\ b & & c \\ d & & e \\ f & & g \\ h & & i \\ j & & k \\ l & & m \\ & p \end{aligned} ,

ȳA = \begin{aligned}& · \\ · & & a \\ b & & b \\ b & & b + d + e \\ b + c + d + f & & b + e + f \\ b + e & & b + d + e + f + g + h + i \\ b + d + e + g + j & & b + e + j \\ b + e + f + i + j & & b + d + e + g + j + l + m \\ & b + e + f + i + j + m + n \end{aligned}

The Uniserial modules of length 1:

A^{(1)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & 1 \end{aligned} , x̄A^{(1)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & · \end{aligned} , ȳA^{(1)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & · \end{aligned}

A_{r}^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & r \\ & · \end{aligned} ,  x̄A_{r}^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & r \end{aligned} ,  ȳA_{r}^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & 1 \end{aligned}

B^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ & · \end{aligned} ,  x̄B^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & 1 \end{aligned} ,  ȳB^{(2)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & · \end{aligned}

The Uniserial modules of length 3:

A_{s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & 1 \\ · & & s \\ & · \end{aligned} ,  x̄A_{s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & 1 \\ & s \end{aligned} ,  ȳA_{s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ & 1 \end{aligned}

B_{s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ & · \end{aligned} ,  x̄B_{s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ & · \end{aligned} ,  ȳB_{s}^{(3)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ & 1 + s \end{aligned}

A_{s,t}^{(4)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & 1 \\ · & & s \\ · & & t \\ & · \end{aligned} ,  x̄A_{s,t}^{(4)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & 1 \\ · & & s \\ & t \end{aligned} ,  ȳA_{s,t}^{(4)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & 1 \\ 1 & & 1 + s \\ & 1 + s \end{aligned}

B_{s,t}^{(4)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ t & & · \\ & · \end{aligned} ,  x̄B_{s,t}^{(4)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ & · \end{aligned} ,  ȳB_{s,t}^{(4)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & s \\ & t \end{aligned}

The Uniserial modules of length 5:

A_{s,u}^{(5)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & 1 \\ · & & s \\ · & & 1 \\ · & & u \\ & · \end{aligned} ,  x̄A_{s,u}^{(5)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & 1 \\ · & & s \\ · & & 1 \\ & u \end{aligned} ,  ȳA_{s,u}^{(5)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & 1 \\ & · \end{aligned}

B_{s,u}^{(5)} = ; \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ s^{2} + s + 1 & & · \\ u & & · \\ & · \end{aligned} ,  x̄B_{s,u}^{(5)} = ; \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ s^{2} + s + 1 & & · \\ & · \end{aligned} ,  ȳB_{s,u}^{(5)} = ; \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & s \\ 1 + s & & s^{2} + 1 \\ & 1 + s + u \end{aligned}

A_{s,u,v}^{(6)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ 1 & & 1 \\ · & & s \\ · & & 1 \\ · & & u \\ · & & v \\ & · \end{aligned} ,  x̄A_{s,u,v}^{(6)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & 1 \\ · & & s \\ · & & 1 \\ · & & u \\ & v \end{aligned} ,  ȳA_{s,u,v}^{(6)} =\begin{matrix}· \\ · & · \\ · & · \\ · & · \\ 1 & 1 \\ · & s \\ 1 & · \\ 1+s & 1+u \\ 1+s+u \end{matrix}

B_{s,u,v}^{(6)} = ; \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ s^{2} + s + 1 & & · \\ u & & · \\ v & & · \\ & · \end{aligned} , x̄B_{s,u,v}^{(6)} = ; \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ s^{2} + s + 1 & & · \\ u & & · \\ & · \end{aligned} , ȳB_{s,u,v}^{(6)} = ; \begin{matrix}· \\ · & · \\ · & · \\ · & · \\ · & · \\ · & 1 + s \\ s^{2} + s & s^{2} + s + 1 \\ s^{2} + s + 1 & s^{2} + s + u \\ s^{2} + s + 1 + v \end{matrix}

The Uniserial modules of length 7:

A_{s,u,w}^{(7)} = \begin{aligned}& · \\ · & & · \\ 1 & & 1 \\ · & & s \\ · & & 1 \\ · & & u \\ · & & s^{2} + s + 1 + u \\ · & & w \\ & · \end{aligned} ,

x̄A_{s,u,w}^{(7)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ 1 & & 1 \\ · & & s \\ · & & 1 \\ · & & u \\ · & & s^{2} + s + 1 + u \\ & w \end{aligned} ,  ȳA_{s,u,w}^{(7)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & · \\ 1 & & 1 \\ 1 & & s + 1 \\ s & & 1 \\ · & & s^{2} + 1 + u \\ & s^{2} + s + 1 + u \end{aligned}

B_{s,u,w}^{(7)} = \begin{aligned}& · \\ · & & · \\ · & & 1 \\ s & & · \\ s^{2} + s + 1 & & · \\ u & & · \\ 1 + s^{2} + s^{3} + s^{4} & & · \\ w & & · \\ & · \end{aligned} ,  x̄B_{s,u,w}^{(7)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & · \\ s^{2} + s + 1 & & · \\ u & & · \\ 1 + s^{2} + s^{3} + s^{4} & & · \\ & · \end{aligned} ,

ȳB_{s,u,w}^{(7)} = \begin{aligned}& · \\ · & & · \\ · & & · \\ · & & 1 \\ s & & 1 + s \\ 1 & & s^{2} \\ 1 + u & & 1 + u \\ 1 + s + u & & u + s^{2} + s^{3} + s^{4} \\ & 1 + s + u + w \end{aligned}

There are no uniserial modules of length 8.

§7 Uniserial submodules of 𝔽D_{8}Y4

Group: D_{8}Y4 = ⟨x, y, z | x^{4} = y^{2} = z^{2} = 1, x ∈ Z, (yz)^{2} = x^{2}⟩, where Z is the center of the group.

This is the partially ordered set of uniserial 𝔽D_{8}Y4 submodules:

[figure: A_{r,s}^{(2)}; B_{r}^{(2)}; C^{(2)}; A^{(1)}]

This is the homomorphism diagram for uniserial 𝔽D_{8}Y4 modules:

[figure: \underline{A_{r,s}^{(2)}}; \underline{B_{r}^{(2)}}; \underline{C^{(2)}}; \underline{A^{(1)}}]

As there are no uniserial modules of a length greater than 2, all uniserial modules are mutually non-isomorphic.

A basis for 𝔽D_{8}Y4 is:

1

x̄  ȳ  z̄
x̄^{2}  x̄ȳ  x̄z̄  ȳz̄
x̄^{3}  x̄^{2}ȳ x̄^{2}z̄  x̄ȳz̄
x̄^{3}ȳ  x̄^{3}z̄  x̄^{2}ȳz̄
x̄^{3}ȳz̄

A = \begin{aligned}& a \\ b & & c & & d \\ e & & f & g & & h \\ i & & j & k & & l \\ & m & & n & & p \\ & & & & q \end{aligned} ,  x̄A = \begin{aligned}& · \\ a & & · & & · \\ b & & c & d & & · \\ e & & f & g & & h \\ & j & & k & & l \\ & & & & p \end{aligned} ,

ȳA = \begin{aligned}& · \\ · & & a & & · \\ · & & b & · & & d \\ · & & e & · & & g \\ & i & & · & & k \\ & & & & n \end{aligned}

z̄A = \begin{aligned}& · \\ · & & · & & a \\ c & & · & b & & c \\ f & & c & c + e + h & & f \\ & f & & f + i + l & & c + h + j \\ & & & & f + l + m \end{aligned}

The Uniserial module of length 1:

A^{(1)} = \begin{aligned}& · \\ · & & · & & · \\ · & & · & · & & · \\ · & & · & · & & · \\ & · & & · & & · \\ & & & & 1 \end{aligned} ,  x̄A^{(1)} = \begin{aligned}& · \\ · & & · & & · \\ · & & · & · & & · \\ · & & · & · & & · \\ & · & & · & & · \\ & & & & · \end{aligned} ,

ȳA^{(1)} = \begin{aligned}& · \\ · & & · & & · \\ · & & · & · & & · \\ · & & · & · & & · \\ & · & & · & & · \\ & & & & · \end{aligned}  z̄A^{(1)} = \begin{aligned}& · \\ · & & · & & · \\ · & & · & · & & · \\ · & & · & · & & · \\ & · & & · & & · \\ & & & & · \end{aligned}

The Uniserial modules of length 2:

A_{r,s}^{(2)} = \begin{aligned}& · \\ · & & · & & · \\ · & & · & · & & · \\ · & & · & · & & · \\ & r & & s & & 1 \\ & & & & · \end{aligned} ,  x̄A_{r,s}^{(2)} = \begin{aligned}& · \\ · & & · & & · \\ · & & · & · & & · \\ · & & · & · & & · \\ & · & & · & & · \\ & & & & 1 \end{aligned} ,

ȳA_{r,s}^{(2)} = \begin{aligned}& · \\ · & & · & & · \\ · & & · & · & & · \\ · & & · & · & & · \\ & · & & · & & · \\ & & & & s \end{aligned}  z̄A_{r,s}^{(2)} = \begin{aligned}& · \\ · & & · & & · \\ · & & · & · & & · \\ · & & · & · & & · \\ & · & & · & & · \\ & & & & r \end{aligned}

B_{r}^{(2)} = \begin{aligned}& · \\ · & & · & & · \\ · & & · & · & & · \\ · & & · & · & & · \\ & 1 & & r & & · \\ & & & & · \end{aligned} ,  x̄B_{r}^{(2)} = \begin{aligned}& · \\ · & & · & & · \\ · & & · & · & & · \\ · & & · & · & & · \\ & · & & · & & · \\ & & & & · \end{aligned} ,
ȳB_{r}^{(2)} = \begin{aligned}& · \\ · & & · & & · \\ · & & · & · & & · \\ · & & · & · & & · \\ & · & & · & & · \\ & & & & r \end{aligned}  z̄B_{r}^{(2)} = \begin{aligned}& · \\ · & & · & & · \\ · & & · & · & & · \\ · & & · & · & & · \\ & · & & · & & · \\ & & & & 1 \end{aligned}
C^{(2)} = \begin{aligned}& · \\ · & & · & & · \\ · & & · & · & & · \\ · & & · & · & & · \\ & · & & 1 & & · \\ & & & & · \end{aligned} ,  x̄C^{(2)} = \begin{aligned}& · \\ · & & · & & · \\ · & & · & · & & · \\ · & & · & · & & · \\ & · & & · & & · \\ & & & & · \end{aligned} ,
ȳC^{(2)} = \begin{aligned}& · \\ · & & · & & · \\ · & & · & · & & · \\ · & & · & · & & · \\ & · & & · & & · \\ & & & & 1 \end{aligned} ,  z̄C^{(2)} = \begin{aligned}& · \\ · & & · & & · \\ · & & · & · & & · \\ · & & · & · & & · \\ & · & & · & & · \\ & & & & · \end{aligned}

There are no uniserial modules of length 3.

Chapter 3: Tensor decompositions of the Regular Representation

In this chapter we present several theorems related to tensor decompositions of group rings, culminating in a description of tensor decompositions of the group rings of particular groups of order 16. Throughout, 𝔽 is a field of characteristic p, where p is a prime. Unless otherwise noted, G is a p-group, and H and K are subgroups of G. We define M and N to be left 𝔽G-modules, and we denote the Loewy length of the module M by ℓℓ(M).

The tensor product M ⊗ N is the set of 𝔽-linear combinations of the elements m ⊗_{𝔽} n, where m ∈ M and n ∈ N. The group G acts on M ⊗ N by g · m ⊗ n = gm ⊗ gn, for any element g of G. Extending 𝔽-linearly gives M ⊗ N a left 𝔽Gmodule structure. To distinguish multiplication of m ⊗ n by g from multiplication of m by g, we will write g · m ⊗ n and gm respectively.

Two of the more important results of this section deal with the particular situation where 𝔽G = M ⊗ N, and ℓℓ(𝔽G) = ℓℓ(M) + ℓℓ(N) − 1. Under these circumstances, Theorem 1.6 states that the radical and socle series of M coincide, as do those of N. In Theorem 1.7 we see that the quotient of consecutive powers of the radical of 𝔽G may be written entirely in terms of M and N.

We note here that Theorems 1.5 through 1.7, and Corollary 1.8 have immediate generalizations, without significantly changing the proofs. We may replace the supposition that 𝔽G ≅ M ⊗ N with the supposition that M ⊗ N is a module for which the radical series and the socle series coincide. The conjecture after Corollary 1.8 may likewise be broadened in a number of ways.

§1 Tensor Decompositions

In the following two lemmas we do not require G to be a p-group.

Lemma 1.1. If g is an element of the group G, and m and n are elements of left 𝔽G-modules, then

(g − 1) · m ⊗ n = (g − 1)m ⊗ n + m ⊗ (g − 1)n + (g − 1)m ⊗ (g − 1)n.

Proof. Care must be taken in noting the differences between the two definitions of multiplication.

(g − 1) · m ⊗ n = g · m ⊗ n − m ⊗ n = gm ⊗ gn − m ⊗ n
= gm ⊗ gn − m ⊗ n − gm ⊗ n + gm ⊗ n

= gm ⊗ (g − 1)n + (g − 1)m ⊗ n − m ⊗ (g − 1)n + m ⊗ (g − 1)n

= (g − 1)m ⊗ (g − 1)n + (g − 1)m ⊗ n + m ⊗ (g − 1)n

Lemma 1.2. The product of the augmentation ideal and M ⊗N, IG·M ⊗N, is contained in IG M ⊗ N + M ⊗ IG N.

Proof. An element of M ⊗ N is a linear combination of elements of the form m⊗n, where m and n are elements of some fixed bases for M and N, respectively. A basis for the augmentation ideal IG is the set {g −1 | 1 ≠ g ∈ G}, so IG·M ⊗N consists of linear combinations of elements of the form (g − 1) · m ⊗ n. From the previous lemma, (g −1)·m⊗n = (g −1)m⊗n+m⊗(g −1)n+(g −1)m⊗(g −1)n, which is the sum of elements of IG M ⊗ N, M ⊗ IG N, and IG M ⊗ IG N. Since IG M ⊗ IG N is contained in both IG M ⊗ N and M ⊗ IG N, the lemma is proved.

If we again require G to be a p-group, then the radical and the augmentation ideal coincide. The following results are obtained.

Lemma 1.3. If M and N are left 𝔽G-modules, then

1) Rad^{k}(M ⊗ N) ⊆ ∑_{i=0}^{k} Rad^{i}(M) ⊗ Rad^{k−i}(N)

2) Soc^{k}(M ⊗ N) ⊇ ∑_{i=0}^{k} Soc^{i}(M) ⊗ Soc^{k+1−i}(N)

Proof. For statement 1), when k = 0 we have equality. Assume that statement 1) is true for all k ≤ K. Then

Rad^{K+1}(M ⊗ N) = IG · Rad^{K}(M ⊗ N)

⊆ ∑_{i=0}^{K} IG · Rad^{i}(M) ⊗ Rad^{K−i}(N)

⊆ ∑_{i=0}^{K} Rad^{i+1}(M) ⊗ Rad^{K−i}(N) + Rad^{i}(M) ⊗ Rad^{K+1−i}(N)

= ∑_{i=0}^{K+1} Rad^{i}(M) ⊗ Rad^{K+1−i}(N).

As for statement 2), it is evidently true when k = 0. Assume that statement 2) is true for all k < K. With k = K, we work with the radical of any summand of the right side.

Rad(Soc^{i}(M) ⊗ Soc^{K+1−i}(N)) = IG · Soc^{i}(M) ⊗ Soc^{K+1−i}(N) ⊆ IG Soc^{i}(M) ⊗ Soc^{K+1−i}(N) + Soc^{i}(M) ⊗ IG Soc^{K+1−i}(N)

⊆ Soc^{i−1}(M) ⊗ Soc^{K+1−i}(N) + Soc^{i}(M) ⊗ Soc^{K−i}(N)
⊆ Soc^{K−1}(M ⊗ N).

Since the radical of Soc^{i}(M) ⊗ Soc^{K+1−i}(N) is contained in Soc^{K−1}(M ⊗ N), we see that Soc^{i}(M) ⊗ Soc^{K+1−i}(N) is contained in Soc^{K}(M ⊗ N).

Lemma 1.4. The Loewy length of the tensor product of 𝔽G-modules M ⊗ N has an upper bound, ℓℓ(M ⊗ N) ≤ ℓℓ(M) + ℓℓ(N) − 1.

Proof. The Loewy length of a module M is defined as the minimum power k such that Rad^{k}(M) = 0. Using the previous lemma, any element of Rad^{k}(M ⊗N) is a sum of elements in the modules Rad^{i}(M) ⊗ Rad^{k−i}(N). To obtain a non-zero element in the latter tensor product we must have i < ℓℓ(M) and k − i < ℓℓ(N). Thus ℓℓ(M ⊗ N) − 1 ≤ ℓℓ(M) − 1 + ℓℓ(N) − 1.

Theorem 1.5. Assume 𝔽G ≅ M ⊗ N. Then ℓℓ(𝔽G) = ℓℓ(M) + ℓℓ(N) − 1 if and only if the inclusions of Lemma 1.3 become equalities.

Proof. Let λ = ℓℓ(𝔽G), μ = ℓℓ(M), and ν = ℓℓ(N). By Lemma 1.4, λ ≤ μ + ν − 1. As G is a p-group, the radical and socle series of 𝔽G coincide. In other words, Soc^{i}(𝔽G) = Rad^{λ−i}(𝔽G). Using this, Lemma 1.3 implies

∑_{i=0}^{k} Soc^{i}(M) ⊗ Soc^{k+1−i}(N) ⊆ Soc^{k}(𝔽G) = Rad^{λ−k}(𝔽G)

⊆ ∑_{j=0}^{λ−k} Rad^{λ−k−j}(M) ⊗ Rad^{j}(N),

(∗)

for any k between 0 and λ. As Rad^{μ}(M) = 0, and Rad^{ν}(N) = 0, we know that

Rad^{λ−k−j}(M) ⊗ Rad^{j}(N) ⊆ Soc^{μ+k+j−λ}(M) ⊗ Soc^{ν−j}(N).  (∗∗)

First, we assume that λ = μ + ν − 1. Inclusion (∗∗) becomes

Rad^{μ+ν−1−k−j}(M) ⊗ Rad^{j}(N) ⊆ Soc^{j+k+1−ν}(M) ⊗ Soc^{ν−j}(N),

which, when combined with the inclusions (∗) gives

∑_{i=0}^{k} Soc^{i}(M) ⊗ Soc^{k+1−i}(N) ⊆ ∑_{j=0}^{μ+ν−1−k} Rad^{μ+ν−1−k−j}(M) ⊗ Rad^{j}(N)

⊆ ∑_{j=0}^{μ+ν−1−k} Soc^{j+k+1−ν}(M) ⊗ Soc^{ν−j}(N).

Replacing j with i + ν − k − 1, the last module becomes

∑_{i=k+1−ν}^{μ} Soc^{i}(M) ⊗ Soc^{k+1−i}(N).

We know that Soc^{k}(M) = 0 if k ≤ 0, and Soc^{k}(M) = M if k ≥ ℓℓ(M), for any module M. Thus we have

∑_{i=0}^{k} Soc^{i}(M) ⊗ Soc^{k+1−i}(N) = ∑_{i=k+1−ν}^{μ} Soc^{i}(M) ⊗ Soc^{k+1−i}(N),

which implies that the inclusions of (∗) are equalities.

Now we assume that the set inclusions of (∗) are equalities. With k = 0 we obtain

0 = Rad^{λ}(𝔽G) = ∑_{j=0}^{λ} Rad^{λ−j}(M) ⊗ Rad^{j}(N).

But if λ − j ≤ μ − 1 and j ≤ ν − 1, the summation on the right can not be zero. Thus μ − 1 + ν − 1 ≤ λ − 1. But we know from Lemma 1.4 that λ ≤ μ + ν − 1, so we must have equality.

Remark: We see from the proof of Theorem 1.5 that equality in condition 1) of Lemma 1.3, when k = λ by itself is enough to imply that ℓℓ(𝔽G) = ℓℓ(M) + ℓℓ(N) − 1, which in turn implies equality in conditions 1) and 2) of Lemma 1.3. Thus, if 𝔽G ≅ M ⊗ N, and 0 = Rad^{λ}(M ⊗ N) = ∑_{j=0}^{λ} Rad^{λ−j}(M) ⊗ Rad^{j}(N), where λ = ℓℓ(M ⊗ N), then ℓℓ(𝔽G) = ℓℓ(M) + ℓℓ(N) − 1, and equality holds in the inclusions of Lemma 1.3.

Theorem 1.6. If 𝔽G ≅ M ⊗ N, and ℓℓ(𝔽G) = ℓℓ(M) + ℓℓ(N) − 1, then the socle series and radical series of M coincide, as do those of N.

Proof. We are given that 𝔽G ≅ M ⊗ N. Let μ = ℓℓ(M), ν = ℓℓ(N), and λ = ℓℓ(𝔽G) = μ + ν − 1. By Theorem 1.5 we have

Soc^{1}(M) ⊗ Soc^{1}(N) = Soc^{1}(M ⊗ N)

= Rad^{λ−1}(M ⊗ N) = ∑_{j=0}^{λ−1} Rad^{λ−1−j}(M) ⊗ Rad^{j}(N).

As Rad^{ν}(N) = 0, any summand with j ≥ ν is zero. Also, as Rad^{μ}(M) = 0, the non-zero terms in the sum occur when μ − 1 ≥ λ − 1 − j = μ + ν − 2 − j, which gives j ≥ ν − 1. Thus the only non-zero term in this sum occurs when j = ν − 1, and we have

Soc^{1}(M) ⊗ Soc^{1}(N) = Rad^{μ−1}(M) ⊗ Rad^{ν−1}(N).

We must have Soc^{1}(M) = Rad^{μ−1}(M) and Soc^{1}(N) = Rad^{ν−1}(N).

Assume that Soc^{k}(M) = Rad^{μ−k}(M) and Soc^{k}(N) = Rad^{ν−k}(N), for all k < K. We know that

Soc^{K}(M) ⊗ Soc^{1}(N) ⊆ ∑_{j=0}^{μ+ν−1−K} Rad^{μ+ν−1−K−j}(M) ⊗ Rad^{j}(N).

Each summand with j ≥ ν is zero. The summand with j = ν − 1 is

Rad^{μ−K}(M) ⊗ Rad^{ν−1}(N) ⊆ Soc^{K}(M) ⊗ Soc^{1}(N).  (†)

The summands with values of j less than ν − 1 are contained in the module

Rad^{μ−K+1}(M) ⊗ Rad^{0}(N) = Soc^{K−1}(M) ⊗ N.

Given m ⊗ n ∈ Soc^{K}(M) ⊗ Soc^{1}(N), we may write m ⊗ n = a + b, where a ∈ Rad^{μ−K}(M)⊗Soc^{1}(N), and b ∈ Soc^{K−1}(M)⊗N. As a ∈ Soc^{K}(M)⊗Soc^{1}(N), b = m ⊗ n − a must also be an element of Soc^{K}(M) ⊗ Soc^{1}(N). Thus, as these are tensor products of vector spaces,

b ∈ (Soc^{K−1}(M) ⊗ N) ∩ (Soc^{K}(M) ⊗ Soc^{1}(N)) = Soc^{K−1}(M) ⊗ Soc^{1}(N)
= Rad^{μ+1−K}(M) ⊗ Rad^{ν−1}(N) ⊆ Rad^{μ−K}(M) ⊗ Rad^{ν−1}(N).

Thus m⊗n ∈ Rad^{μ−K}(M)⊗Soc^{1}(N), for all elements m⊗n of Soc^{K}(M)⊗Soc^{1}(N), and the inclusion of (†) is reversed, so

Soc^{K}(M) ⊗ Soc^{1}(N) = Rad^{μ−K}(M) ⊗ Rad^{ν−1}(N).

From this we deduce that Soc^{K}(M) = Rad^{μ−K}(M). By a symmetric argument, Soc^{K}(N) = Rad^{μ−K}(N) for all K.

Theorem 1.7. If 𝔽G ≅ M ⊗ N and ℓℓ(𝔽G) = ℓℓ(M) + ℓℓ(N) − 1, then

Rad^{k}(𝔽G)/ Rad^{k+1}(𝔽G) ≅

⨁_{i=0}^{k} ( Rad^{i}(M)/ Rad^{i+1}(M)) ⊗ ( Rad^{k−i}(N)/ Rad^{k−i+1}(N)).

Proof. We start by creating a homomorphism between the modules. We represent elements of Rad^{i}(M) and Rad^{k−i}(N) by m_{i} and n_{k−i} respectively. Let ϕ be the map from the direct sum on the right to Rad^{k}(M ⊗ N)/ Rad^{k+1}(M ⊗ N) defined on each term of each summand by

ϕ ((m_{i} + Rad^{i+1}(M)) ⊗ (n_{k−i} + Rad^{k−i+1}(N))) = m_{i} ⊗n_{k−i} +Rad^{k+1}(M ⊗N).

We extended ϕ 𝔽-linearly to the entire sum. This map is clearly a well defined 𝔽G-homomorphism. Every element of Rad^{k}(M ⊗ N) is the sum of elements of the form m_{i} ⊗ n_{k−i} by Theorem 1.5, so ϕ is surjective.

We know that the dimensions of the quotients Rad^{k}(𝔽G)/ Rad^{k+1}(𝔽G) must add up to dim(𝔽G) = |G|. Let r_{i} = dim (Rad^{i}(M)/ Rad^{i+1}(M)), and let s_{j} = dim (Rad^{j}(N)/ Rad^{j+1}(N)). Let L = ℓℓ(𝔽G), μ = ℓℓ(M), and ν = ℓℓ(N). We have

∑_{k=0}^{L} dim (⨁_{i=0}^{k} ( Rad^{i}(M)/ Rad^{i+1}(M)) ⊗ ( Rad^{k−i}(N)/ Rad^{k−i+1}(N))) =

∑_{k=0}^{L} ∑_{i=0}^{k}(r_{i}s_{k−i}) = ∑_{i=0}^{μ} ∑_{j=0}^{ν} r_{i}s_{i} = dim(M) dim(N) = |G|.

The homomorphism ϕ is a surjection of one module to another where both modules have the same dimension. This implies that ϕ is an isomorphism.

For an 𝔽G-module M, the Poincaré polynomial associated to the filtration of M by its radical powers is defined as the polynomial

P_{M}(t) = ∑_{i=0}^{∞} t^{i} dim(Rad^{i}(M)/ Rad^{i+1}(M).

We note that if P_{M}(t) is a polynomial of degree k, then there is no power of t less than k for which the coefficient is 0. In terms of Poincaré polynomials, Theorem 1.7 may be restated as,

Corollary 1.8. If 𝔽G ≅ M ⊗ N and ℓℓ(𝔽G) = ℓℓ(M) + ℓℓ(N) − 1, then P_{𝔽G}(t) = P_{M}(t)P_{N}(t).

Conjecture. If 𝔽G ≅ M ⊗ N, then ℓℓ(𝔽G) = ℓℓ(M) + ℓℓ(N) − 1.

This conjecture holds true for all cases where |G| ≤ 16.

Lemma 1.9. Let M and N be left 𝔽G-modules. The following statements are equivalent:

1) M ⊗ N ≅ 𝔽G

2) dim(M ⊗ N) = |G| and M ⊗ N may be generated by a single element,

3) dim(M ⊗ N) = |G| and ∥G∥ · M ⊗ N ≠ 0,

4) dim(M ⊗ N) = |G| and ℓℓ(M ⊗ N) = ℓℓ(𝔽G).

Proof. We start by showing that 2), 3) and 4) follow immediately from 1). Assume that M ⊗ N ≅ 𝔽G. The dimension of 𝔽G is |G|, and as M ⊗ N ≅ 𝔽G, dim(M ⊗ N) = |G|. Let ϕ be an isomorphism, ϕ : 𝔽G → M ⊗ N. Since 1 generates 𝔽G as a left 𝔽G-module, the element ϕ(1) also generates M ⊗ N as a left 𝔽G-module. The product ∥G∥ · 1 = ∥G∥ ≠ 0, so likewise ∥G∥ · ϕ(1) ≠ 0. Isomorphic modules have the same Loewy length.

Assume that dim(M ⊗ N) = |G|, and that there is an element a which generates M ⊗ N. Let ϕ be the left 𝔽G-module homomorphism ϕ : 𝔽G → M ⊗ N given by ϕ(x) = x · a. As ϕ(𝔽G) = M ⊗ N, and dim(M ⊗ N) = dim(𝔽G), the homomorphism must be an isomorphism. Thus condition 2) implies condition 1).

Assume that dim(M ⊗N) = |G|, and that ∥G∥·M ⊗N ≠ 0. This implies that there must exist an element m⊗n of M ⊗N for which ∥G∥·m⊗n ≠ 0. Let ϕ be the left 𝔽G-module homomorphism ϕ : 𝔽G → M ⊗ N given by ϕ(x) = x · m ⊗ n. Then ϕ(∥G∥) = ∥G∥ · m ⊗ n ≠ 0. Since G is a p-group, the socle of 𝔽G is generated by the element ∥G∥. The kernel of ϕ must be 0, as any larger kernel would intersect the socle non-trivially. As the dimensions of 𝔽G and M ⊗ N are the same, the homomorphism ϕ must be an isomorphism. Thus condition 3) implies condition 1).

Assume that dim(M ⊗ N) = |G|, and that ℓℓ(M ⊗ N) = ℓℓ(𝔽G). Let l = ℓℓ(M ⊗N). We have Rad^{l}(𝔽G)·M ⊗N = 0, but Rad^{l−1}(𝔽G)·M ⊗N ≠ 0. Since G is a p-group, the radical and socle series coincide, and Rad^{l−1}(𝔽G) = Soc(𝔽G) = 𝔽∥G∥. Thus condition 4) implies condition 3).

Note: Conditions 1) and 2) are equivalent even if G is not a p-group.

We now observe certain characteristics of possible factors in a tensor decomposition.

Lemma 1.10. If 𝔽G ≅ M ⊗ N then both M and N are cyclic modules.

Proof. We assume that 𝔽G ≅ M ⊗ N. By Lemma 1.9, there must be an element m ⊗ n which generates M ⊗ N. Since the dimensions of 𝔽G and M ⊗ N is |G|, the set {g · m ⊗ n = gm ⊗ gn | g ∈ G} forms a basis for M ⊗ N. This set is contained in the set {gm ⊗ hn | g, h ∈ G}. The modules spanned by {gm|g ∈ G} and {hn|h ∈ G} form submodules of M and N, respectively. The dimensions of modules are related by the inequalities,

dim(𝔽G) = |G| = |{gm ⊗ gn | g ∈ G}|
≤ dim(Span({gm ⊗ hn | g, h ∈ G}))
= dim(Span({gm|g ∈ G})) dim(Span({hn|h ∈ G}))
≤ dim(M) dim(N) = dim(M ⊗ N) = dim(𝔽G).

We must have equality throughout, and M and N must be generated by m and n respectively.

We recall the following notation from the introduction.

For a p-group G, Jennings [J, Al 1, Sc] describes a decreasing series of subgroups, κ_{i}(G) = {g ∈ G|g ≡ 1 modulo Rad^{i}(𝔽G)}, which we refer to as the Jennings series of G. This series of subgroups has the following properties:

1) [κ_{λ}, κ_{μ}] ⊆ κ_{λ+μ},

2) g^{p} ∈ κ_{ip} for all g ∈ κ_{i},

3) κ_{λ}/κ_{2λ} is elementary abelian. For each i ≥ 1 choose elements x_{i,s} of G such that the set {x_{i,s}κ_{i+1} | 1 ≤

s ≤ d_{i}\} forms a basis for κ_{i}/κ_{i+1}. Let \overline{x}_{i,s} = x_{i,s} − 1 ∈ 𝔽G. There are |G| products of the form ∏ \overline{x}_{i,s}^{α_{i,s}}, where the factors are listed in lexicographic order, and 0 ≤ α_{i,s} ≤ p − 1. The weight of such a product is defined to be ∑ iα_{i,s}. Jennings’ theorem states that the set of products of weight w lie in Rad^{w}(𝔽G), and form a basis modulo Rad^{w+1}(G).

Alperin comments [Al 1] that the order of the factors is irrelevant. After choosing a particular order for these factors, let {β_{i,t}} be the set of such products with weight i. For a product β_{j,t} = ∏ \overline{x}_{i,s}^{α_{i,s}} Alperin defines a complementary element, β_{j,t}^{c} = ∏ \overline{x}_{i,s}^{p−1−α_{i,s}}. We further define a coefficient c_{i,t} = ∏ \binom{p−1}{α_{i,s}}. The element β_{λ−1,1} = ∏ \overline{x}_{i,s}^{p−1} = ∥G∥ is the generator of the socle of 𝔽G, and has weight λ − 1 = ℓℓ(𝔽G) − 1. The sum of the weights of β_{i,t} and β_{i,t}^{c} is λ − 1.

The following result will be used in section 3 when we determine the tensor decomposition of 𝔽G by uniserial modules.

Lemma 1.11. With the notation just defined, if M and N are left 𝔽Gmodules, m ∈ Soc^{μ}(M), n ∈ Soc^{ν}(N), and μ + ν − 1 = λ = ℓℓ(𝔽G), then

∥G∥ · m ⊗ n = ∑_{t} c_{μ−1,t}β_{μ−1,t}m ⊗ β_{μ−1,t}^{c}n.

Proof. We use the fact that ∥G∥ = ∏ \overline{x}_{i,s}^{p−1}, where we have specified the order of multiplication. We start by multiplying m ⊗ n by the last term in the product. We have \overline{x}_{i,s} · m ⊗ n = (x_{i,s} − 1) · m ⊗ n = \overline{x}_{i,s}m ⊗ n + m ⊗ \overline{x}_{i,s}n + \overline{x}_{i,s}m ⊗ \overline{x}_{i,s}n. Since \overline{x}_{i,s} ∈ Rad^{i}(𝔽G), we have \overline{x}_{i,s}m ∈ Soc^{μ−i}(M) and \overline{x}_{i,s}n ∈ Soc^{ν−i}(N).

We proceed by multiplying by \overline{x}_{i,s}^{p−1}. At each stage, we have a sum of elements which is multiplied on the left by \overline{x}_{i,s}. Each element gives a sum of three further elements, where \overline{x}_{i,s} is applied on the left tensor factor, the right factor, or both factors. Thus

\overline{x}_{i,s}^{p−1} · m ⊗ n = S_{1} + ∑_{j=0}^{p−1} \binom{p − 1}{j}\overline{x}_{i,s}^{j}m ⊗ \overline{x}_{i,s}^{p−1−j}n,

where S_{1} is the sum of all elements for which the multiplication was applied to both tensor factors at least once.

We continue this process for the entire product, and obtain

∥G∥ · m ⊗ n = S + ∑_{γ,t} c_{γ,t}β_{γ,t}m ⊗ β_{γ,t}^{c}n,

where S is the sum of all elements for which at least on of the factors of ∥G∥ was applied to both tensor factors at the same stage.

Now m is an element of Soc^{μ}(M). Thus if m is multiplied by an element of Rad^{μ}(𝔽G), the result is zero. Likewise, if n is multiplied by an element of Rad^{ν}(𝔽G), the result is zero. The weights of β_{γ,t} and β_{γ,t}^{c} are γ and λ − γ − 1, respectively. The sum of these weights is λ − 1 = μ − 1 + ν − 1. The weight of an element is the minimum power of the radical in which it lies. Thus S must be zero, as well as all other summands except for the terms c_{μ−1,t}β_{μ−1,t}m ⊗ β_{μ−1,t}^{c}n.

Lemma 1.12. If m and n are elements of 𝔽G, and Rad(𝔽Gn) ⊆ Rad^{2}(𝔽Gm) then Rad(𝔽Gm) = Rad(𝔽G(m + n)).

Proof. Given g ∈ G, we know that \overline{g}n = (g − 1)n is an element of Rad(𝔽Gn). Let a_{1} = −\overline{g}. As \overline{g}n ∈ Rad^{2}(𝔽Gm), there is an element a_{2} ∈ Rad^{2}(𝔽G) such that a_{2}m = −a_{1}n = \overline{g}n. We find elements a_{i} ∈ Rad^{i}(𝔽G) such that a_{i+1}m = −a_{i}n. We set a_{i} = 0 for all i ≥ μ = ℓℓ(𝔽Gm). We find that

∑_{i=2}^{μ} a_{i}(m + n) = ∑_{i=2}^{μ} a_{i}n − ∑_{i=1}^{μ−1} a_{i}n = a_{μ}n − a_{1}n = \overline{g}n,

which is an element of Rad^{2}(𝔽G(m + n)). Thus Rad^{1}(𝔽Gn) ⊆ Rad^{2}(𝔽G(m + n)). The submodule Rad^{1}(𝔽G(m+n)) is generated by elements of the form (g−1)(m+ n) = \overline{g}(m + n) = \overline{g}m + \overline{g}n. But \overline{g}n ∈ Rad^{2}(𝔽G(m + n), so the radical is generated by all elements of the form \overline{g}m. This is also the radical of 𝔽Gm.

If we have one tensor decomposition of 𝔽G, we now show a way to find other tensor decompositions.

Lemma 1.13. If m and m_{1} are elements of 𝔽G, Rad(𝔽Gm_{1}) ⊆ Rad^{2}(𝔽Gm), 𝔽G ≅ 𝔽Gm⊗N, and ℓℓ(𝔽Gm)+ℓℓ(N)−1 = ℓℓ(𝔽G) then 𝔽G ≅ 𝔽G(m+m_{1})⊗N.

Proof. By Lemma 1.10 we see that there is an element n that generates N. Lemma 1.9 implies that there is an element m′ ⊗ n′ such that ∥G∥m′ ⊗ n′ ≠ 0. As m and n generate 𝔽Gm and N, we may assume that m′ = m+rm and n′ = n+sn, where r and s are elements of Rad(𝔽G), as non-zero scalar multiples are irrelevant.

Let μ = ℓℓ(𝔽Gm), and ν = ℓℓ(N). We must have m and m′ in Soc^{μ}(𝔽Gm) and n and n′ in Soc^{ν}(N). By Lemma 1.11 we see that

∥G∥ · m ⊗ n = ∑_{t} c_{μ−1,t}β_{μ−1,t}m ⊗ β_{μ−1,t}^{c}n.

Replacing m and n by m′ and n′ respectively, we see that β_{μ−1,t}(m + rm) = β_{μ−1,t}m, and β_{μ−1,t}^{c}(n + sn) = β_{μ−1,t}^{c}n, as rm and sn are not in a high enough power of the socle to survive. Thus we may assume that m′ = m, and n′ = n. Any choice of generators for the two modules would suffice.

We now substitute m + m_{1} for m. We have β_{μ−1,t}(m + m_{1}) = β_{μ−1,t}m + β_{μ−1,t}m_{1}. But m_{1} ∈ Soc^{μ−1}(𝔽G), so this term is just β_{μ−1,t}m. We have ∥G∥ · (m + m_{1}) ⊗ n = ∥G∥ · m ⊗ n ≠ 0. By Lemma 1.12, we have Rad(𝔽Gm) = Rad(𝔽G(m+m_{1})). Since these modules are both cyclic, and have the same radical, they must both have the same dimension. Again, using Lemma 1.9, we see that 𝔽G ≅ 𝔽G(m + m_{1}) ⊗ N.

We next present a result relating the radical series to the commutativity within 𝔽G. This result will be useful to us in specific calculations involving the action of generators of Rad(𝔽G) on modules.

For the following lemma, G need not be a p-group.

Lemma 1.14. If G is a group, H is a subgroup of index p, and g is an element of G not in H, then ∥G∥ = (g − 1)^{p−1}∥H∥ in a field of characteristic p.

Proof. As g is outside of H, the cosets of H in G are {g^{i}H | 0 ≤ i < p}. The element (g − 1)^{p} = g^{p} − 1 in 𝔽_{p}G, so (g − 1)^{p−1} is the sum of all powers less than p of g. Thus (g − 1)^{p−1}∥H∥ is the sum of all elements of all cosets of ∥H∥ in G.

Corollary 1.15. For the p-group G with Jennings series G = κ_{1} ⊇ κ_{2} ⊇ …, and the set {x_{i,s} | 1 ≤ i < n, 1 ≤ s ≤ d_{i}} as described above, we may write

∥κ_{i}∥ = ∏_{s=1}^{d_{i}} x̄_{i,s}^{p−1}∥κ_{i+1}∥,

and ∥κ_{i+1}∥ is in the center of 𝔽κ_{i}.

Proof. The group κ_{i}/κ_{i+1} is an elementary abelian group with generators {x_{i.s}κ_{i+1} | 1 ≤ s ≤ d_{i}}. We may thus use the lemma iteratively to obtain the first result. Also, we know that κ_{i+1} is normal in κ_{i}, so

x_{i,s}∥κ_{i+1}∥ = ∥x_{i,s}κ_{i+1}∥ = ∥κ_{i+1}x_{i,s}| = ∥κ_{i+1}∥x_{i,s}.

Corollary 1.16. The center of 𝔽G contains Soc^{2}(𝔽G).

Proof. From the previous corollary we have ∥κ_{2}∥ is contained in the center of 𝔽G. As κ_{2} is the Frattini subgroup of G, we know that

gh∥κ_{2}∥ = hg g^{−}1h^{−}1gh∥κ_{2}∥ = hg∥κ_{2}∥,

since the Frattini subgroup contains the commutator subgroup. By induction, we find that ∥κ_{l}∥ = ∏ x̄_{i,s}^{α_{i,s}}, where the product is taken over all i and s, and α_{i,s} = 0 when i < l, and p − 1 when i ≥ l. Thus ∥κ_{2}∥ = ∏ x̄_{i,s}^{α_{i,s}}, where α_{1,s} = 0 for all s, and α_{i,s} = p − 1 for all i ≥ 2 and all s.

A basis for Soc^{2}(𝔽G) is the set of elements {β_{j,t} | j ≥ ℓℓ(𝔽G) − 2}. Let β_{j,t} = ∏ x̄_{i,s}^{α_{i,s}} be an element of weight j = ℓℓ(𝔽G) − 2. In order to have weight j, we need α_{i,s} = p − 1 except for one exponent, α_{1,t} = p − 2, for some t. We see that ∥κ_{2}∥ is a factor of β_{j,t}, and we may rewrite

β_{j,t} = ∏_{s} x̄_{1,s}^{p−1−δ_{s,t}} ∥κ_{2}∥,

where δ_{s,t} = 1 when s = t, and 0 otherwise. We know that a full set of generators for G is {x_{s} = x̄_{s}+1}. As elements commute modulo ∥κ_{2}∥, we see that gβ_{j,t} = β_{j,t}g for all g in G, and since the β_{j,t} form a basis for Soc^{2}(𝔽G), this establishes the result.

For each maximal subgroup H of G, choose an element g_{H} of G that is not in H. Let S_{H} = ∑_{i=0}^{p−1} ig_{H}^{i}∥H∥. We recall that 𝔽_{p} is the field of p elements.

Theorem 1.17. The complete list of elements of Soc^{2}(𝔽_{p}G) is

{f∥G∥ | f ∈ 𝔽_{p}} ∪ {fg_{H}^{i}S_{H} | H is maximal in G, 0 ≠ f ∈ 𝔽, 0 ≤ i < p}.

Proof. First we show that the elements in question are indeed elements of Soc^{2}(𝔽_{p}G). We know that Soc(𝔽_{p}G) is spanned by ∥G∥. Thus the first set of elements is contained in Soc(𝔽_{p}G). Let H be a maximal subgroup of G. As G is a p-group, every maximal subgroup of G is normal with index p. Thus any element of G may be written in the form g_{H}^{j}h for some element h ∈ H, and some j between 0 and p − 1. As G/H ≅ C_{p}, we have g_{H}^{j}hg_{H}^{i}∥H∥ = g_{H}^{i+j}∥H∥, and

g_{H}^{j}hS_{H} = ∑_{i=0}^{p−1} ig_{H}^{i+j}∥H∥

= ∑_{i=0}^{p−1}(i + j)g_{H}^{i+j}∥H∥ − j ∑_{i=0}^{p−1} g_{H}^{i+j}∥H∥

= S_{H} − j∥G∥.

Thus (g_{H}^{j}h − 1)S_{H} = −j∥G∥ ∈ Soc(𝔽_{p}G). As Rad(𝔽_{p}G) is generated by elements of the form g − 1, we see that S_{H} ∈ Soc^{2}(𝔽_{p}G), for every maximal subgroup H of G. As Soc^{2}(𝔽_{p}G) is an ideal, we also have fg_{H}^{i}S_{H} ∈ Soc^{2}(𝔽_{p}G), for all f ∈ 𝔽_{p}, and 0 ≤ i < p.

We show that no element is listed more than once in the statement of the theorem. Clearly the elements of the first set are distinct, and different from all elements of the second set. Assume that the element f_{1}g_{H}^{i}S_{H} = f_{2}g_{K}^{j}S_{K} for some elements f_{1} and f_{2} of 𝔽_{p}, and some maximal subgroups H and K. In f_{1}g_{H}^{i}S_{H}, every element of a coset gH has the same coefficient. Thus the subgroup H may be determined. This indicates that H = K. Only one of the cosets of H has coefficient 0 in f_{1}g_{H}^{i}S_{H}. This coset must be g_{H}^{i}, as ∥H∥ is the coset in S_{H} with coefficient 0. This forces i and j to be equal. The two expressions of the same element both are the same multiple of the element g_{H}^{i}S_{H}, so f_{1} = f_{2}. Thus the elements of the second set are distinctly listed.

We then count the number of elements of Soc^{2}(𝔽_{p}G), and find it to be the same as the number of elements listed in the statement of the theorem. We know that the Frattini subgroup of G is κ_{2}(G). The elementary abelian group G/κ_{2} is isomorphic to a vector space of dimension d_{1}. The maximal subgroups of G are generated by maximal subspaces of G/κ_{2}. The number of maximal subspaces of a vector space is equal to the number of projective lines in that subspace. The number of projective lines in G/κ_{2} is (p^{d_{1}} − 1)/(p − 1). Thus, the number of non-zero elements of the form fg_{H}^{i}S_{H} is (p − 1)p(p^{d_{1}} − 1)/(p − 1) = p^{d_{1}+1} − p. The number of elements of Soc(𝔽_{p}G) is p. Thus, the number of elements listed is p^{d_{1}+1}.

The vector space Soc^{2}(𝔽_{p}G) has a basis {x̄_{1,s}^{c} | 1 ≤ s ≤ d_{1}} ∪ {∥G∥}. This gives dim(Soc^{2}(𝔽_{p}G)) = d_{1} + 1, and the number of elements in Soc^{2}(𝔽_{p}G) is p^{d_{1}+1}.

The particular case where p = 2 has a simpler form.

Corollary 1.18. If G is a 2-group then the elements of Soc^{2}(𝔽_{2}G) consists of the elements

{0, ∥G∥} ∪ {∥H∥, ∥H∥^{c} | H is maximal in G},

where ∥H∥^{c} refers to the complement of ∥H∥, i.e. ∥G∥ − ∥H∥.

Proof. The field 𝔽_{2} has only 2 elements, 1 and 0. Given a maximal subgroup H of G, let g be an element of G not in H. We see that S_{H} = g∥H∥, and gS_{H} = ∥H∥. In the statement of the theorem, we see that f and i may only be 0 or 1, and the result follows.

§2 Permutation Module Decompositions

In this section, we look at decompositions of 𝔽G into the tensor product M ⊗ N, where both M and N are permutation modules. By restricting our attention to permutation modules, we are able to obtain cleaner results than appear to hold for general tensor decompositions. A permutation module for 𝔽G is a module which has a basis permuted by G. The first lemma is well known.

Lemma 2.1. Let M be a left 𝔽G permutation module with G acting transitively on the basis B of M. If H = Stab_{G}(b), for some basis element b ∈ B, then M ≅ 𝔽G∥H∥ ≅ 𝔽[G/H].

Proof. Let T be a left transversal for the subgroup H in G. Let 1 represent the coset 1H. The basis B for M is the G orbit of b, i.e., B = {gb | g ∈ G} = {tb | t ∈ T }. Likewise, a basis for 𝔽G∥H∥ is {t∥H∥ = ∥tH∥ | t ∈ T }, and a basis for 𝔽[G/H] is {tH | t ∈ T }. Make the relation between basis elements,

tb ↔ t∥H∥ ↔ tH.

This identification preserves the action of G, and the isomorphism is found by extending linearly to the entirety of the modules.

Theorem 2.2. Let H and K be subgroups of G and let M and N be the left 𝔽G-modules generated by ∥H∥ and ∥K∥, respectively. Then 𝔽G ≅ M ⊗ N if and only if G = HK and H ∩ K = 1.

Proof. We give a proof in the case where G is a p-group and 𝔽 has characteristic p. If HK ≠ G then either H ∩ K ≠ 1, or |H||K| < |G|. If |H||K| < |G|, then the dimension of 𝔽G∥H∥ ⊗ 𝔽G∥K∥ is |H||K|, and is less than |G|, which is the dimension of 𝔽G.

Let Q = H ∩ K. The element ∥H∥ ⊗ ∥K∥ is a generator of 𝔽G∥H∥ ⊗ 𝔽G∥K∥. We have,

∥Q∥ · ∥H∥ ⊗ ∥K∥ = ∑_{q∈Q} q∥H∥ ⊗ q∥K∥

= ∑_{q∈Q} ∥H∥ ⊗ ∥K∥

= |Q|∥H∥ ⊗ ∥K∥.

If Q ≠ 1, then this product is equal to 0, as p divides the order of Q. However, ∥Q∥1 = ∥Q∥ ≠ 0 in 𝔽G. If an element of 𝔽G other than zero annihilates a generator of a module, then that module must not be isomorphic to the regular representation.

Bases for M and N are {∥kH∥ : k ∈ K} and {∥hK∥} : h ∈ H}, respectively. (Each basis element is the sum of elements of a particular coset of the group.) A basis for M ⊗ N is then

{∥kH∥ ⊗ ∥hK∥ : k ∈ K, h ∈ H}.

The group G permutes these basis elements. The total number of basis elements is |G|. The stabilizer of the element ∥H∥ ⊗ ∥K∥ must be in both H and K, so Stab_{G}(∥H∥ ⊗ ∥K∥) = 1. This shows that the size of the orbit of ∥H∥ ⊗ ∥K∥ is |G|, so the orbit must be the entire basis. The basis is permuted regularly by G, so M ⊗ N ≅ 𝔽G.

Corollary 2.3. Let 𝔽 be a field of characteristic p. Let G be a group, which is not necessarily a p-group. Let H and K be subgroups of G with trivial intersection. Let M and N be the 𝔽G-modules generated by ∥H∥ and ∥K∥, respectively. Then 𝔽G ≅ M ⊗ N if and only if G = HK.

Proof. The only part of the proof of the theorem which uses the fact that G is a p-group involves the order of the intersection of H and K. If we insist that this intersection is trivial, the rest of the proof holds.

We comment that an analogous, but more complicated statement may be made concerning decompositions of 𝔽G as a tensor product of three or more permutation modules.

Let H and K be subgroups of the group G. Then K is a complement of H if HK = G, and H ∩ K = 1.

If α is an automorphism of the group G, and K is a complement to H, it is a simple matter to show that α(K) is also a complement of α(H) in G. It is useful to know that if α is an inner automorphism, then H also has the complement α(K).

Lemma 2.4. Let H and K be subgroups in G. If K is a complement of H, and g is any element of G, then ^{g}K is also a complement of H.

Proof. We may write g = hk for some elements h of H and k of K. Now H ^{g}K = HhkKk^{−1}h^{−1} = HKh^{−1} = Gh^{−1} = G. Since |H ∩ ^{g}K||H ^{g}K| = |H|| ^{g}K| = |G|, we have |H ∩ ^{g}K| = 1.

§3 Tensor Decompositions of 𝔽G, where |G| = 16

In this paper we have been paying particular attention to the groups of order 16 which may not be non-trivially written as the direct product of two subgroups. The calculations we are about to perform depend heavily on an examination of the subgroup lattices of these groups, which we take to be readily available for inspection either from [H&S] or by calculations with the program [Sc]. We also rely on the lists of uniserial modules for each group ring, which were obtained in Chapter 2.

For each group G of order 16, we consider non-trivial tensor decompositions 𝔽G ≅ M ⊗ N. If 𝔽G ≅ M ⊗ N, then dim(M) dim(N) = 16 and ℓℓ(𝔽G) ≤ ℓℓ(M) + ℓℓ(N) − 1 by Lemma 1.4. By symmetry of the tensor decomposition, we need only consider cases where dim(M) ≤ dim(N), so we must have dim(M) = 2 and dim(N) = 8 or dim(M) = dim(N) = 4. By Lemma 1.10 we need only consider cyclic modules M and N. If M is cyclic and dim(M) = 2, then M is uniserial, and ℓℓ(M) = 2.

We first consider tensor decompositions in which both factors are permutation modules. If such a decomposition exists, Theorem 2.2 tells us that there must be a pair of complementary subgroups H and K in G for which M is generated by ∥H∥ and N is generated by ∥K∥. By symmetry of the tensor product, we may take H to be the larger of the two subgroups. The dimension of the module is equal to the index of the subgroup in the group, so |H| = 8 and |K| = 2, or |H| = |K| = 4. With these choices, G = HK only when H ∩ K = 1, so we need only consider pairs which have trivial intersections. By Lemma 2.4 we need only consider representatives of conjugacy classes for both H and K.

For each group the full subgroup lattice is given. The larger subgroups are listed above the smaller subgroups. Each subgroup of a given order is listed at the same level. A solid line between two subgroups indicates that the smaller subgroup is contained in the larger subgroup, and there is no intermediate subgroup. The diagram also indicates the orbits of subgroups under the actions of the automorphisms of the group. A triple dotted line between two entries indicates that the two subgroups are conjugates. A single dotted line between the listings of two subgroups indicates that there is an outer automorphism of G that maps one subgroup to the other. If there is an automorphism that maps one subgroup to another subgroup, then there is a path of dotted lines connecting the listings of the two subgroups.

We start with the groups G of order 16 for which 𝔽G has Loewy length 7. These are the groups G_{1} and (2 × 4).2. In both cases, the Poincaré polynomial associated to 𝔽G is P_{𝔽G}(t) = 1 + 2t + 3t^{2} + 4t^{3} + 3t^{4} + 2t^{5} + t^{6}. Assume 𝔽G ≅ M ⊗ N. If dim(M) = 2 then 7 ≤ 2 + ℓℓ(N) − 1 ≤ 1 + 6 = 7, as ℓℓ(N) < ℓℓ(𝔽G). Thus ℓℓ(N) = 6 and dim(N) = 8. The Poincaré polynomials for M and N are P_{M}(t) = 1+t and P_{N}(t) = 1+t+2t^{2} +2t^{3} +t^{4} +t^{5}. If dim(M) = dim(N) = 4 then 7 ≤ ℓℓ(M)+ℓℓ(N)−1 ≤ 4+4−1 = 7, forcing equality, and both M and N must be uniserial. Their corresponding Poincaré polynomials are both P (t) = 1+t+t^{2}+t^{3}.

Given M and N, we wish to determine if 𝔽G ≅ M ⊗ N. We use Lemma 1.9 to determine this. We only consider cases where dim(M) dim(N) = 16, and either ℓℓ(M) = 2 and ℓℓ(N) = 6 or ℓℓ(M) = ℓℓ(N) = 4. If we can show that the product ∥G∥ · M ⊗ N ≠ 0, then we know that 𝔽G ≅ M ⊗ N. Let m and n be generators of M and N respectively. By Lemma 1.11 of Chapter 1, the choice of generators does not matter. The product ∥G∥ · m ⊗ n is non-zero if and only if the product ∥G∥ · M ⊗ N is non-zero. We use Lemma 1.11 to find ∥G∥ · m ⊗ n.

In both groups that we are considering, we have chosen bases for 𝔽G consisting of elements of the form β_{i+j,i} = x̄^{i}ȳ^{j}, where 0 ≤ i, j ≤ 3. The complementary element is β_{i+j,i}^{c} = x̄^{3−i}ȳ^{3−j}. Note that the characteristic of 𝔽 is 2, so c_{i+j,i} = 1.

If dim(M) = 2 and dim(N) = 8, we have μ = 2 and

\begin{aligned}∥G∥ · m ⊗ n & = ∑_{t} c_{μ−1,t}β_{μ−1,t}m ⊗ β_{μ−1,t}^{c}n \\ & = x̄m ⊗ x̄^{2}ȳ^{3}n + ȳm ⊗ x̄^{3}ȳ^{2}n. \end{aligned}

If dim(M) = dim(N) = 4, we have μ = 4 and

∥G∥ · m ⊗ n = x̄^{3}m ⊗ ȳ^{3}n + x̄^{2}ȳm ⊗ x̄ȳ^{2}n + x̄ȳ^{2}m ⊗ x̄^{2}ȳn + ȳ^{3}m ⊗ x̄^{3}n.

We now focus our attention on the group

G_{1} = ⟨x, y | x^{4} = y^{4} = (xy)^{2} = 1, {x^{2}, y^{2}} ⊂ Z⟩ ,

where Z is the center of the group. We first consider the decomposition of this group into the non-trivial product of subgroups H and K whose intersection is 1.

We present the diagram for the full subgroup lattice of G_{1}:

Theorem 3.1. The group ring 𝔽G_{1} is decomposable into the tensor product of permutation modules, 𝔽G∥H∥ ⊗ 𝔽G∥K∥. Up to interchanging H and K, these are the only such pairs of subgroups.

a) The subgroup H is either ⟨x, y^{2}⟩ or ⟨x^{2}, y⟩ and K is any one of the non-central subgroups of order 2;

b) The subgroup H is either ⟨x⟩ or ⟨xy^{2}⟩ and K is one of the four subgroups ⟨xy, x^{2}y^{2}⟩, ⟨x^{−1}y, x^{2}y^{2}⟩, ⟨xy, y^{2}⟩, or ⟨x^{−1}y, y^{2}⟩;

b′) The subgroup H is either ⟨y⟩ or ⟨x^{2}y⟩ and K is one of the four subgroups

⟨xy, x^{2}y^{2}⟩, ⟨x^{−1}y, x^{2}y^{2}⟩, ⟨xy, x^{2}⟩, or ⟨xy^{−1}, x^{2}⟩;

c) The subgroup H is either ⟨x⟩ or ⟨xy^{2}⟩ and K is either ⟨y⟩ or ⟨x^{2}y⟩.

Note that the complementary pairs of b′) may be obtained by applying the automorphism that exchanges x and y to the complementary pairs of b).

Proof. By observing the diagram, we determine all complementary pairs of subgroups in G_{1}. By Theorem 2.2 we see that these subgroup pairs correspond with all possible tensor decompositions of 𝔽G_{1} by permutation modules.

We now list, up to isomorphism, all permutation modules of 𝔽G_{1}, identifying those that are uniserial on our list of all uniserial modules. To every subgroup H of G_{1}, there is a module 𝔽G_{1}∥H∥. We list the subgroups in the order of decreasing size, with conjugate subgroups listed together. For each subgroup H, we list a generator of 𝔽G_{1}∥H∥, as described in Chapter 2, section 1. The generators of the form E_{r,s,t,u,v}^{(6)} are described after the listing.

• ⟨x, y⟩ = G_{1} corresponds to A^{(1)}.

• ⟨x^{2}, y^{2}, xy⟩ ≅ C_{2}^{3} corresponds to A_{1}^{(2)}.

• ⟨y, x^{2}⟩ ≅ C_{2} × C_{4} corresponds to A_{0}^{(2)}.

• ⟨x, y^{2}⟩ ≅ C_{2} × C_{4} corresponds to B^{(2)}.

• ⟨xy, x^{2}y^{2}⟩ ≅ C_{2}^{2} corresponds to A_{1,1,1}^{(4)} = C_{1,1,0,1}^{(4)}.

• ⟨x^{−1}y, x^{2}y^{2}⟩ ≅ C_{2}^{2} corresponds to A_{1,0,0}^{(4)} = C_{1,0,0,0}^{(4)}.

• C_{2}^{2} ≅ ⟨xy, x^{2}⟩ and ⟨xy^{−1}, x^{2}⟩ correspond to D_{0,0,0}^{(4)} and D_{1,0,1}^{(4)}.

• C_{2}^{2} ≅ ⟨xy, y^{2}⟩ and ⟨x^{−1}y, y^{2}⟩ correspond to C_{0,1,0,1}^{(4)} and C_{0,0,0,0}^{(4)}.

• ⟨x^{2}, y^{2}⟩ ≅ C_{2}^{2} corresponds to x̄^{2}ȳ^{2}, which is not a uniserial module generator.

• C_{4} ≅ ⟨x⟩ and ⟨xy^{2}⟩ correspond to B_{1,0}^{(4)} and B_{0,0}^{(4)}.

• C_{4} ≅ ⟨y⟩ and ⟨x^{2}y⟩ correspond to A_{0,0,0}^{(4)} and A_{0,1,0}^{(4)}.

• C_{2} ≅ ⟨xy⟩ and ⟨x^{−1}y^{−1}⟩ correspond to E_{1,0,0,0,0}^{(6)} and E_{0,1,0,0,0}^{(6)}.

• C_{2} ≅ ⟨xy^{−1}⟩ and ⟨x^{−1}y⟩ correspond to E_{1,1,0,0,0}^{(6)} and E_{0,0,0,0,0}^{(6)}.

• ⟨x^{2}⟩ ≅ C_{2} corresponds to x̄^{2}, which is not a uniserial module generator.

• ⟨y^{2}⟩ ≅ C_{2} corresponds to ȳ^{2}, which is not a uniserial module generator.

• ⟨x^{2}y^{2}⟩ ≅ C_{2} corresponds to x̄^{2}+ȳ^{2}, which is not a uniserial module generator.

• ⟨1⟩ generates all of 𝔽G_{1}.

We now find all modules M and N such that 𝔽G_{1} ≅ M ⊗N. We start with the case where dim(M) = 2 and dim(N) = 8. We need to find all cyclic submodules N of 𝔽G_{1} with Poincaré polynomial P_{N} = 1 + t + 2t^{2} + 2t^{3} + t^{4} + t^{5}. We use an algorithm similar to that described in Chapter 1, Section 2. We sketch the use of the algorithm, although the actual calculations are performed by computer, and are not shown here. We use the basis from Chapter 2, Section 1 for 𝔽G_{1}. We start with an arbitrary element of Soc^{6}(𝔽G_{1}),

E^{(6)} = \begin{aligned}& & & · \\ & & b & & c \\ & d & & e & & f \\ g & & h & & i & & j \\ & k & & l & & m \\ & & n & & p \\ & & & q \end{aligned} .

Let N = 𝔽G_{1}E^{(6)}. We need E^{(6)} ∉ Soc^{5}(𝔽G_{1}). This implies that b and c can not both be zero. Consider the map from Rad(N) to Rad(N)/ Rad^{2}(N), where we list the elements in terms of the images of the basis elements x̄^{2}, x̄ȳ, and ȳ^{2} of 𝔽G_{1}. Using the multiplications that are given in Chapter 2, Section 1, we find that the images of x̄E^{(6)} and ȳE^{(6)} are (b, c, 0) and (b, b, b + c) respectively. Since we need dim(Rad(N)/ Rad^{2}(N)) = 1, we see that both b and c must be equal. By rescaling, we choose b = c = 1.

We find the set of elements x̄^{i}ȳ^{j}E^{(6)}, and perform a Gram-Schmidt type of process on the set. For each of the entries b, d, g, i, k, l, n, and q in the original arbitrary element E^{(6)}, we find an element in our resulting set that has a 1 in the corresponding position, and only zeros in all prior entries. We then conclude that a distinguished generator of our module should have a zero in each of these positions, other than the position designated by b. After setting the values of these variables as indicated, we find that there remain two vectors that are not in the form of a distinguished generator. The module is either an 8-dimensional or a 10-dimensional vector space, depending on the values of these two vectors. As we desire an 8-dimensional space, we require these two remaining vectors to be zero. The two vectors are zero precisely when j = e + ef + e^{2} + f + h. We thus set j to this value. The distinguished generators of 𝔽G_{1} which generate 8-dimensional modules of Loewy length 6 are of the form

E_{r,s,t,u,v}^{(6)} = \begin{aligned}& & & · \\ & & 1 & & 1 \\ & · & & r & & s \\ · & & t & & · & & r + rs + r^{2} + s + t \\ & · & & · & & u \\ & & · & & v \\ & & & · \end{aligned}

The isomorphism bucket for the module generated by such an element is

\underline{E_{r,s,t,u,v}^{(6)}_{r+s=α,r^{2}t+v=β}} ; α, β, t ∈ 𝔽

In order to find all modules M and N such that 𝔽G_{1} ≅ M ⊗ N, we need to find the products ∥G_{1}∥ · m ⊗ n that are non-zero, where m generates M, and n generates N. As we have seen, if M is two dimensional, and N is eight dimensional, then we need to find x̄m, ȳm, x̄^{2}ȳ^{3}n, and x̄^{3}ȳ^{2}n. We present these products in the following multiplication table:

E_{r,s,t,u,v}^{(6)} A_{r}^{(2)} B^{(2)}
x̄^{2}ȳ^{3} A^{(1)}  0  0
x̄^{3}ȳ^{2} A^{(1)}  0  0
x̄  −  A^{(1)} 0
ȳ  −  rA^{(1)} A^{(1)}

Table of elements of (IG_{1})u and (IG_{1})^{5}u

Any entry of − in this table is an irrelevant product. We see that

\begin{aligned}∥G_{1}∥A_{ρ}^{(2)} ⊗ E_{r,s,t,u,v}^{(6)} & = x̄A_{ρ}^{(2)} ⊗ x̄^{2}ȳ^{3}E_{r,s,t,u,v}^{(6)} + ȳA_{ρ}^{(2)} ⊗ x̄^{3}ȳ^{2}E_{r,s,t,u,v}^{(6)} \\ & = (1 + ρ)A^{(1)} ⊗ A^{(1)}, \end{aligned}

and this is non-zero whenever ρ ≠ 1. Likewise, if we substitute B^{(2)} in the previous equation in place of A^{(2)}, we find that ∥G_{1}∥B^{(2)} ⊗ E_{r,s,t,u,v}^{(6)} = A^{(1)} ⊗ A^{(1)} ≠ 0. We have found the tensor decompositions 𝔽G_{1} ≅ M ⊗ N where dim(M) = 2 and dim(N) = 8. We restate these results in the following table. We list the possible values of m on the top, and the possible values of n on the side. The coefficient of A^{(1)} ⊗ A^{(1)} in the product ∥G_{1}∥ · m ⊗ n is listed in the corresponding row and column. Any non-zero entry indicates that there is a tensor decomposition 𝔽G_{1} ≅ 𝔽G_{1}m ⊗ 𝔽G_{1}n.

A_{ρ}^{(2)} B^{(2)}
E_{r,s,t,u,v}^{(6)} 1 + ρ 1

Table of coefficients for A^{(1)} ⊗ A^{(1)} in ∥G_{1}∥ · m ⊗ n If M and N are both 4-dimensional, we need to find the products, x̄^{3}u, x̄^{2}ȳu,

x̄ȳ^{2}u, and ȳ^{3}u, where u is a generator of M or N. We have seen that both M and N need to be uniserial. We obtain the following multiplication table:

A_{r,s,t}^{(4)} B_{s,t}^{(4)} C_{s,t,u,v}^{(4)} D_{s,t,u}^{(4)}
x̄^{3} A^{(1)} 0 A^{(1)}  0
x̄^{2}ȳ rA^{(1)} 0 A^{(1)}  0
x̄ȳ^{2} rA^{(1)} 0 sA^{(1)} A^{(1)}
ȳ^{3} r^{2}A^{(1)} A^{(1)} sA^{(1)} A^{(1)}

Table of elements of (IG_{1})^{3} u We wish to determine all non-zero products of the form ∥G_{1}∥ · m ⊗ n = x̄^{3}m ⊗ ȳ^{3}n + x̄^{2}ȳm ⊗ x̄ȳ^{2}n + x̄ȳ^{2}m ⊗ x̄^{2}ȳn + ȳ^{3}m ⊗ x̄^{3}n.

This table gives the coefficient of A^{(1)} ⊗ A^{(1)} in the product ∥G_{1}∥ · m ⊗ n.

A_{r,s,t}^{(4)}  B_{s,t}^{(4)}  C_{s,t,u,v}^{(4)}  D_{s,t,u}^{(4)}
A_{ρ,σ,τ}^{(4)}  r^{2} + ρ^{2}  1 (s + ρ)(ρ + 1) 1 + ρ
B_{σ,τ}^{(4)}  1  0  1  0
C_{σ,τ,υ,ν}^{(4)} (σ + r)(r + 1) 1  0  0
D_{σ,τ,υ}^{(4)}  1 + r  0  0  0

Table of coefficients of A^{(1)} ⊗ A^{(1)} in ∥G∥ · m ⊗ n

This table describes all tensor decompositions of 𝔽G_{1} ≅ M ⊗ N, when both M and N are uniserial. A generator of each uniserial module of length 4 is listed on the top and on the side. If the entry corresponding to generators m and n is non-zero, then a tensor decomposition by those modules exists. For example, the entry corresponding to A_{r,s,t}^{(4)} and A_{ρ,σ,τ}^{(4)} is r^{2} + ρ^{2}. This indicates that if r^{2} + ρ^{2} ≠ 0, (i.e., r ≠ ρ), then 𝔽G_{1} ≅ 𝔽G_{1}A_{ρ,σ,τ}^{(4)} ⊗ 𝔽G_{1}A_{r,s,t}^{(4)}. In this manner, we determine all tensor decompositions of 𝔽G_{1}:

Theorem 3.2. Up to isomorphism and rearrangement of factors, the following is the complete list of non-trivial tensor decompositions, 𝔽G_{1} ≅ M ⊗ N:

a) M ≅ 𝔽G_{1}A_{ρ}^{(2)} and N ≅ 𝔽G_{1}E_{0,α,t,0,β}^{(6)} when ρ ≠ 1;

b) M ≅ 𝔽G_{1}B^{(2)} and N ≅ 𝔽G_{1}E_{0,α,t,0,β}^{(6)};

c) M ≅ 𝔽G_{1}A_{ρ,σ,τ}^{(4)} and N ≅ 𝔽G_{1}A_{r,s,t}^{(4)} when ρ ≠ r;

d) M ≅ 𝔽G_{1}A_{ρ,σ,τ}^{(4)} and N ≅ 𝔽G_{1}B_{s,t}^{(4)};

e) M ≅ 𝔽G_{1}A_{ρ,σ,τ}^{(4)} and N ≅ 𝔽G_{1}C_{s,t,u,v}^{(4)} when s ≠ ρ and ρ ≠ 1, and if s = 1 then u ≠ 0;

f) M ≅ 𝔽G_{1}A_{ρ,σ,τ}^{(4)} and N ≅ 𝔽G_{1}D_{s,t,u}^{(4)} with ρ ≠ 1;

g) M ≅ 𝔽G_{1}B_{σ,τ}^{(4)} and N ≅ 𝔽G_{1}C_{s,t,u,v}^{(4)} such that if s = 1 then u ≠ 0.

The restrictions on C_{s,t,u,v}^{(4)} preventing s = 1 and u = 0 are to prevent overlap in the listing, as C_{1,t,0,v}^{(4)} = A_{1,t,v}^{(4)}. For the modules of the form 𝔽G_{1}E_{r,s,t,u,v}^{(6)} only one element was chosen to represent the isomorphism class. The same was not done with the modules of Loewy length 4.

For each of the remaining groups, we present similar diagrams and tables. In the tables listing group ring elements of (IG)^{i}u, we list the coefficients of A^{(1)}. In the tables listing ∥G∥ · m ⊗ n, we list the coefficients of A^{(1)} ⊗ A^{(1)}.

We turn our attention to the other group which has Loewy length 7,

(2 × 4).2 = ⟨x, y | x^{4} = y^{4} = 1, ^{x}y = y^{−1}⟩.

We first consider the decomposition of this group into the non-trivial product of subgroups H and K whose intersection is 1. We present the diagram for the full subgroup lattice of (2 × 4).2:

Theorem 3.3. The group ring 𝔽(2×4).2 is decomposable into the tensor product of permutation modules, 𝔽(2 × 4).2∥H∥ ⊗ 𝔽(2 × 4).2∥K∥. Up to interchanging H and K, these are the only such pairs of subgroups:

a) H is one of the four subgroups ⟨x⟩, ⟨xy^{2}⟩, ⟨xy⟩, and ⟨xy^{−1}⟩;

b) K is one of the two subgroups ⟨y⟩ and ⟨x^{2}y⟩ .

Proof. Again, we observe the lattice diagram, and conclude that these are all the possible pairs of complementary subgroups in (2 × 4).2.

We now list, up to isomorphism, all permutation modules of 𝔽(2×4).2, identifying those that are uniserial on our list of all uniserial modules. To every subgroup H of (2×4).2, there is a module 𝔽(2×4).2∥H∥. We list the subgroups in the order of decreasing size, with conjugate subgroups listed together. For each subgroup H, we list a generator of 𝔽(2 × 4).2∥H∥, as described in Chapter 2, section 2.

• ⟨x, y⟩ = (2 × 4).2 corresponds to A^{(1)}.

• ⟨x, y^{2}⟩ ≅ C_{2} × C_{4} corresponds to A_{0}^{(2)}.

• ⟨xy, y^{2}⟩ ≅ C_{2} × C_{4} corresponds to A_{1}^{(2)}.

• ⟨y, x^{2}⟩ ≅ C_{2} × C_{4} corresponds to B^{(2)}.

• ⟨x^{2}, y^{2}⟩ ≅ C_{2}^{2} corresponds to x̄^{2}ȳ^{2}, which is not a uniserial module generator.

• C_{4} ≅ ⟨x⟩ and ⟨xy^{2}⟩ correspond to A_{0,1,0}^{(4)} and A_{0,0,0}^{(4)}.

• C_{4} ≅ ⟨xy⟩ and ⟨xy^{−1}⟩ correspond to A_{1,0,0}^{(4)} and A_{1,1,1}^{(4)}.

• C_{4} ≅ ⟨y⟩ and ⟨x^{2}y⟩ correspond to B_{0,0}^{(4)} and B_{1,0}^{(4)}.

• ⟨x^{2}⟩ ≅ C_{2} corresponds to x̄^{2}, which is not a uniserial module generator.

• ⟨y^{2}⟩ ≅ C_{2} corresponds to ȳ^{2}, which is not a uniserial module generator.

• ⟨x^{2}y^{2}⟩ ≅ C_{2} corresponds to x̄^{2}+ȳ^{2}, which is not a uniserial module generator.

• ⟨1⟩ generates all of 𝔽(2 × 4).2.

We now find all modules M and N such that 𝔽(2 × 4).2 ≅ M ⊗ N. We start with the case where dim(M) = 2 and dim(N) = 8. There is no pair of complementary subgroups H and K in (2 × 4).2 such that |H| = 8 and |K| = 2. Thus there is no pair of permutation modules M and N with dimensions 2 and 8 such that 𝔽(2 × 4).2 ≅ M ⊗ N. We now look for a module N that is not a permutation module, but would suffice as an 8-dimensional factor in a tensor decomposition of 𝔽(2 × 4).2.

We need to find all cyclic submodules N of 𝔽(2×4).2 with Poincaré polynomial P_{N} = 1 + t + 2t^{2} + 2t^{3} + t^{4} + t^{5}. We use the basis from Chapter 2, Section 2 for 𝔽(2 × 4).2. We start with an arbitrary element of Soc^{6}(𝔽(2 × 4).2),

E^{(6)} = \begin{aligned}& & & · \\ & & b & & c \\ & d & & e & & f \\ g & & h & & i & & j \\ & k & & l & & m \\ & & n & & p \\ & & & q \end{aligned} .

Let N = 𝔽(2×4).2E^{(6)}. We need E^{(6)} ∉ Soc^{5}(𝔽(2×4).2), which implies that b and c can not both be zero. Consider the map from Rad(N) to Rad(N)/ Rad^{2}(N), where we list the elements in terms of the images of the basis elements x̄^{2}, x̄ȳ, and ȳ^{2} of 𝔽(2×4).2. Using the multiplications that are given in Chapter 2, Section 2, we find that the images of x̄E^{(6)} and ȳE^{(6)} are (b, c, 0) and (0, b, b+c) respectively. We see that it is impossible to choose b and c such that dim(Rad(N)/ Rad^{2}(N)) = 1. Thus there are no 𝔽(2 × 4).2-modules that have this Poincaré polynomial, and no tensor decomposition 𝔽(2×4).2 ≅ M ⊗N such that dim(M) = 2 and dim(N) = 8.

If 𝔽(2 × 4).2 ≅ M ⊗ N, then both M and N must be uniserial with Loewy length 4. We give a table of all products bu, where b is one of our basis element of Soc^{4}(𝔽(2×4).2) that is not an element of Soc^{3}(𝔽(2×4).2), and u is a distinguished generator of a uniserial module of Loewy length 4. These basis elements and generators are listed in Chapter 2, section 2. We list the coefficients of A^{(1)} in the products.

A_{r,s,t}^{(4)} B_{s,t}^{(4)}
x̄^{3} r^{2}(r + 1) 1
x̄^{2}ȳ r(r + 1) 0
x̄ȳ^{2}  r  0
ȳ^{3}  1  0

Table of elements of (I(2 × 4).2)^{3} u

From this table, we determine all products,

∥G∥ · m ⊗ n = x̄^{3}m ⊗ ȳ^{3}n + x̄^{2}ȳm ⊗ x̄ȳ^{2}n + x̄ȳ^{2}m ⊗ x̄^{2}ȳn + ȳ^{3}m ⊗ x̄^{3}n.

We list the coefficient of A^{(1)} ⊗ A^{(1)} in the product ∥G∥ · m ⊗ n in the following table.

A_{r,s,t}^{(4)}  B_{s,t}^{(4)}
A_{ρ,σ,τ}^{(4)} (r + ρ)^{2}(1 + r + ρ) 1
B_{σ,τ}^{(4)}  1  0

Table of coefficients of A^{(1)} ⊗ A^{(1)} in ∥(2 × 4).2∥ · m ⊗ n

From this table, we determine all tensor decompositions of 𝔽(2 × 4).2:

Theorem 3.4. Up to isomorphism and rearrangement of factors, the following is the complete list of non-trivial tensor decompositions, 𝔽(2 × 4).2 ≅ M ⊗ N: a) M ≅ 𝔽(2 × 4).2A_{ρ,σ,τ}^{(4)} and N ≅ 𝔽(2 × 4).2A_{r,s,t}^{(4)} such that ρ + r is neither 0

nor 1;

b) M ≅ 𝔽(2 × 4).2A_{ρ,σ,τ}^{(4)} and N ≅ 𝔽(2 × 4).2B_{s,t}^{(4)}.

We now focus on the groups G of order 16 for which 𝔽G has Loewy length 9: Mod_{16}, D_{16}, SD_{16}, and Q_{16}. In all four cases, the Poincaré polynomial associated to 𝔽G is P_{𝔽G}(t) = 1+2t+2t^{2}+2t^{3}+2t^{4}+2t^{5}+2t^{6}+2t^{7}+t^{8}. Assume 𝔽G ≅ M⊗N. If we assume dim(M) = dim(N) = 4 then 9 ≤ ℓℓ(M)+ℓℓ(N)−1 ≤ 4+4−1 = 7 < 9, which indicates a contradiction. If we assume dim(M) = 2 then 9 ≤ 2+ℓℓ(N)−1 ≤ 1 + 8 = 9, as ℓℓ(N) ≤ dim(N). Thus ℓℓ(N) = 8 = dim(N), so both N and M are uniserial. The Poincaré polynomials for M and N are P_{M}(t) = 1 + t and P_{N}(t) = 1 + t + t^{2} + t^{3} + t^{4} + t^{5} + t^{6} + t^{7}.

Given M and N, we wish to determine if 𝔽G ≅ M ⊗ N. We only consider cases where dim(M) dim(N) = 16, ℓℓ(M) = 2 and ℓℓ(N) = 8. If we can show that the product ∥G∥ · M ⊗ N ≠ 0, then we know that 𝔽G ≅ M ⊗ N. Let m and n be generators of M and N respectively. By Lemma 1.11 of Chapter 1, the choice of generators does not matter. The product ∥G∥ · m ⊗ n is non-zero if and only if the product ∥G∥ · M ⊗ N is non-zero. We use Lemma 1.11 to find ∥G∥ · m ⊗ n.

For the groups Mod_{16} and SD_{16} we use the identity,

∥G∥ · (m ⊗ n) = x̄^{7}m ⊗ ȳn + x̄^{6}ȳm ⊗ x̄n,

where m generates a module of length 2, and n generates a module of length 8, and where x̄ and ȳ are defined in Sections 3 and 5 of Chapter 2.

We investigate the modular group of order 16,

Mod_{16} = ⟨x, y | x^{8} = y^{2} = 1, ^{y}x = x^{5}⟩.

We first consider the decomposition of this group into the non-trivial product of subgroups H and K whose intersection is 1. We present the diagram for the full subgroup lattice of Mod_{16}:

Theorem 3.5. The group ring 𝔽Mod_{16} is decomposable into the tensor product of permutation modules, 𝔽Mod_{16}∥H∥ ⊗ 𝔽Mod_{16}∥K∥. Up to interchanging H and K, these are the only such pairs of subgroups:

a) H is one of the two subgroups ⟨x⟩ and ⟨xy⟩;

b) K is one of the two subgroups ⟨y⟩ and ⟨x^{4}y⟩ .

Proof. Again, we observe the lattice diagram, and conclude that these are all the possible pairs of complementary subgroups in Mod_{16}.

We now list, up to isomorphism, all permutation modules of 𝔽Mod_{16}, identifying those that are uniserial on our list of all uniserial modules. To every subgroup H of Mod_{16}, there is a module 𝔽Mod_{16}∥H∥. We list the subgroups in the order of decreasing size, with conjugate subgroups listed together. For each subgroup H, we list a generator of 𝔽Mod_{16}∥H∥, as described in Chapter 2, section 3.

• ⟨x, y⟩ = Mod_{16} corresponds to A^{(1)}.

• ⟨x⟩ ≅ C_{8} corresponds to A_{0}^{(2)}.

• ⟨xy⟩ ≅ C_{8} corresponds to A_{1}^{(2)}.

• ⟨y, x^{2}⟩ ≅ C_{2} × C_{4} corresponds to B^{(2)}.

• ⟨x^{2}⟩ ≅ C_{4} corresponds to x̄^{6}, which is not a uniserial module generator.

• ⟨x^{2}y⟩ ≅ C_{4} corresponds to B_{1,0}^{(4)}.

• ⟨x^{4}, y⟩ ≅ C_{2}^{2} corresponds to B_{0,0}^{(4)}.

• ⟨x^{4}⟩ ≅ C_{2} corresponds to x̄^{4}, which is not a uniserial module generator.

• C_{2} ≅ ⟨y⟩ and ⟨x^{4}y⟩ correspond to B_{0,0,0,0}^{(8)} and B_{1,0,0,0}^{(8)}.

• ⟨1⟩ generates all of 𝔽Mod_{16}.

We now find all modules M and N such that 𝔽Mod_{16} ≅ M ⊗ N. We have seen that if 𝔽Mod_{16} ≅ M ⊗ N then, up to exchanging M and N, the modules must both be uniserial with dimensions 2 and 8. We give a table of all products bu, where b is a one of the basis elements and u is a distinguished generator listed in Section 3 of Chapter 3. Where the product is relevant, we list the coefficient of A^{(1)} in the product.

B_{u,v,w,x}^{(8)} A_{r}^{(2)} B^{(2)}
x̄^{7}  1  0  0
x̄^{6}ȳ  0  0  0
x̄  −  r  1
ȳ  −  1  0

Table of coefficients for A^{(1)} in (IMod_{16})^{n}u

From this table we determine all products,

∥Mod_{16}∥ · m ⊗ n = x̄m ⊗ x̄^{6}ȳn + ȳm ⊗ x̄^{7}n.

We list the coefficient of A^{(1)} ⊗A^{(1)} in the product ∥Mod_{16}∥·m⊗n in the following table.

A_{r}^{(2)} B^{(2)}
B_{u,v,w,x}^{(8)} 1 0

Table of coefficients of A^{(1)} ⊗ A^{(1)} in ∥Mod_{16}∥ · m ⊗ n

From this table, we determine all tensor decompositions of 𝔽Mod_{16}:

Theorem 3.6. Up to isomorphism and rearrangement of factors, 𝔽Mod_{16} ≅ M ⊗ N where M ≅ 𝔽Mod_{16}B_{u,v,w,x}^{(8)}, and N ≅ 𝔽Mod_{16}A_{r}^{(2)}.

We investigate the semidihedral group of order 16,

SD_{16} = ⟨x, y | x^{8} = y^{2} = 1, ^{y}x = x^{3}⟩.

We first consider the decomposition of this group into the non-trivial product of subgroups H and K whose intersection is 1. We present the diagram for the full subgroup lattice of SD_{16}:

Theorem 3.7. The group ring 𝔽SD_{16} is decomposable into the tensor product of permutation modules, 𝔽SD_{16}∥H∥ ⊗ 𝔽SD_{16}∥K∥. Up to interchanging H and K, these are the only such pairs of subgroups:

a) H is one of the two subgroups ⟨x⟩ and ⟨x^{2}, xy⟩;

b) K is one of the four conjugate subgroups ⟨y⟩, ⟨x^{4}y⟩, ⟨x^{2}y⟩, and ⟨x^{6}y⟩ .

Proof. Again, we observe the lattice diagram, and conclude that these are all the possible pairs of complementary subgroups in SD_{16}.

We now list, up to isomorphism, all permutation modules of 𝔽SD_{16}, identifying those that are uniserial on our list of all uniserial modules. To every subgroup H of SD_{16}, there is a module 𝔽SD_{16}∥H∥. We list the subgroups in the order of decreasing size, with conjugate subgroups listed together. For each subgroup H, we list a generator of 𝔽SD_{16}∥H∥, as described in Chapter 2, section 5.

• ⟨x, y⟩ = SD_{16} corresponds to A^{(1)}.

• ⟨xy, x^{2}⟩ ≅ Q_{8} corresponds to A_{1}^{(2)}.

• ⟨x⟩ ≅ C_{8} corresponds to A_{0}^{(2)}.

• ⟨y, x^{2}⟩ ≅ D_{8} corresponds to B^{(2)}.

• ⟨x^{2}⟩ ≅ C_{4} corresponds to x̄^{6}, which is not a uniserial module generator.

• C_{4} ≅ ⟨xy⟩ and ⟨x^{3}y⟩ correspond to A_{1,0}^{(4)} and A_{0,0}^{(4)}.

• C_{2}^{2} ≅ ⟨y, x^{4}⟩ and ⟨x^{2}y, x^{4}⟩ correspond to B_{0,0}^{(4)} and B_{1,0}^{(4)}.

• ⟨x^{4}⟩ ≅ C_{2} corresponds to x̄^{4}, which is not a uniserial module generator.

• C_{2} ≅ ⟨y⟩, ⟨x^{4}y⟩, ⟨x^{2}y⟩, and ⟨x^{6}y⟩ correspond to B_{0,0,0,0}^{(8)}, B_{0,1,0,0}^{(8)}, B_{1,1,1,0}^{(8)} and B_{1,0,0,0}^{(8)}.

• ⟨1⟩ generates all of 𝔽SD_{16}.

We now find all modules M and N such that 𝔽SD_{16} ≅ M ⊗ N. We have seen that if 𝔽SD_{16} ≅ M ⊗ N then, up to exchanging M and N, the modules must both be uniserial with dimensions 2 and 8. We give a table of all products bu, where b is a one of the basis elements and u is a distinguished generator listed in Section 3 of Chapter 3. Where the product is relevant, we list the coefficient of A^{(1)} in the product.

B_{s,u,w,x}^{(8)} A_{r}^{(2)} B^{(2)}
x̄  −  r  1
ȳ  −  1  0
x̄^{7}  1  0  0
x̄^{6}ȳ  0  0  0

Table of coefficients for A^{(1)} in (ISD_{16})^{n}u From this table we determine all products,

∥SD_{16}∥ · m ⊗ n = x̄m ⊗ x̄^{6}ȳn + ȳm ⊗ x̄^{7}n.

We list the coefficient of A^{(1)} ⊗ A^{(1)} in the product ∥SD_{16}∥ · m ⊗ n in the following table.

A_{r}^{(2)} B^{(2)}
B_{s,u,w,x}^{(8)} 1  0

Table of coefficients of A^{(1)} ⊗ A^{(1)} in ∥SD_{16}∥ · m⊗n

From this table, we determine all tensor decompositions of 𝔽SD_{16}:

Theorem 3.8. Up to isomorphism and rearrangement of factors, 𝔽SD_{16} ≅ M ⊗ N where M ≅ 𝔽SD_{16}B_{u,v,w,x}^{(8)}, and N ≅ 𝔽SD_{16}A_{r}^{(2)}.

We investigate the quaternion group of order 16,

Q_{16} = ⟨x, y | x^{8} = 1, x^{4} = y^{2}, ^{y}x = x^{−1}⟩.

We first consider the decomposition of this group into the non-trivial product of subgroups H and K whose intersection is 1. We present the diagram for the full subgroup lattice of Q_{16} after the theorem.

Theorem 3.9. The group ring 𝔽Q_{16} is not decomposable into the tensor product of permutation modules, 𝔽Q_{16}∥H∥ ⊗ 𝔽Q_{16}∥K∥.

Proof. We observe the lattice diagram, and conclude that all non-trivial subgroups of Q_{16} intersect in the subgroup ⟨x^{4}⟩. Thus there is no pair of nontrivial complementary subgroups in Q_{16}.

We now list, up to isomorphism, all permutation modules of 𝔽Q_{16}, identifying those that are uniserial on our list of all uniserial modules. To every subgroup H of Q_{16}, there is a module 𝔽Q_{16}∥H∥. We list the subgroups in the order of decreasing size, with conjugate subgroups listed together. For each subgroup H, we list a generator of 𝔽Q_{16}∥H∥, as described in Chapter 2, section 6.

• ⟨x, y⟩ = Q_{16} corresponds to A^{(1)}.

• ⟨x⟩ ≅ C_{8} corresponds to A_{0}^{(2)}.

• ⟨y, x^{2}⟩ ≅ Q_{8} corresponds to B^{(2)}.

• ⟨xy, x^{2}⟩ ≅ Q_{8} corresponds to A_{1}^{(2)}.

• ⟨x^{2}⟩ ≅ C_{4} corresponds to x̄^{6}, which is not a uniserial module generator.

• C_{4} ≅ ⟨y⟩ and ⟨x^{2}y⟩ correspond to B_{0,0}^{(4)} and B_{1,0}^{(4)}.

• C_{4} ≅ ⟨xy⟩ and ⟨x^{3}y⟩ correspond to A_{1,0}^{(4)} and A_{0,0}^{(4)}.

• ⟨x^{4}⟩ ≅ C_{2} corresponds to x̄^{4}, which is not a uniserial module generator.

• ⟨1⟩ generates all of 𝔽Q_{16}.

Theorem 3.10. There are no non-trivial tensor decompositions of 𝔽Q_{16}.

Proof. From Chapter 2, Section 6, we see that 𝔽Q_{16} has no uniserial modules of length 8. Therefore, there is no non-trivial tensor decomposition of 𝔽Q_{16}.

We investigate the dihedral group of order 16,

D_{16} = ⟨x, y, z | x^{2} = y^{2} = (xy)^{8}, z = xy⟩ .

We first consider the decomposition of this group into the non-trivial product of subgroups H and K whose intersection is 1. We present the diagram for the full subgroup lattice of D_{16}:

Theorem 3.11. The group ring 𝔽D_{16} is decomposable into the tensor product of permutation modules, 𝔽D_{16}∥H∥ ⊗ 𝔽D_{16}∥K∥. Up to interchanging H and K, these are the only such pairs of subgroups:

a) H is the subgroup ⟨x, z^{2}⟩ and K is one of the four subgroups ⟨y⟩, ⟨z^{3}x⟩, ⟨zx⟩, or ⟨yz^{2}⟩;

b) H is the subgroup ⟨y, z^{2}⟩ and K is one of the four subgroups ⟨x⟩, ⟨yz^{3}⟩, ⟨yz⟩, or ⟨z^{2}x⟩;

c) H is the subgroup ⟨z⟩ and K is any of the subgroups of order two other than ⟨z^{4}⟩.

Proof. Again, we observe the lattice diagram, and conclude that these are all the possible pairs of complementary subgroups in D_{16}. Note that there is an automorphism of D_{16} which converts the pairs of statement a) into the pairs of statement b).

We now list, up to isomorphism, all permutation modules of 𝔽D_{16}, identifying those that are uniserial on our list of all uniserial modules. We list all subgroups H of D_{16} along with a generator of the module generated by ∥H∥. Subgroups are listed in decreasing order, and they are listed by conjugacy class.

• ⟨x, y⟩ = D_{16} corresponds with A^{(1)}

• ⟨x, z^{2}⟩ ≅ D_{8} corresponds with A_{0}^{(2)}

• ⟨y, z^{2}⟩ ≅ D_{8} corresponds with B^{(2)}

• ⟨z⟩ ≅ C_{8} corresponds with A_{1}^{(2)}

• C_{2} × C_{2} ≅ ⟨x, z^{4}⟩ and ⟨yz, z^{4}⟩ correspond with A_{0,0}^{(4)} and A_{1,0}^{(4)} respectively • C_{2} × C_{2} ≅ ⟨y, z^{4}⟩ and ⟨zx, z^{4}⟩ correspond with B_{0,0}^{(4)} and B_{1,0}^{(4)} respectively • ⟨z^{2}⟩ ≅ C_{4} corresponds with A_{0}^{(3)} + B_{0}^{(3)}, which generates a 4-dimensional

module

• C_{2} ≅ ⟨x⟩, ⟨yz^{3}⟩, ⟨yz⟩, and ⟨z^{2}x⟩ correspond with A_{0,0,0}^{(8)}, A_{0,1,1}^{(8)}, A_{1,0,0}^{(8)} and A_{1,1,1}^{(8)} respectively

• C_{2} ≅ ⟨y⟩, ⟨z^{3}x⟩, ⟨zx⟩, and ⟨yz^{2}⟩ correspond with B_{0,0,0}^{(8)}, B_{0,1,1}^{(8)}, B_{1,0,0}^{(8)} and B_{1,1,1}^{(8)} respectively

• C_{2} ≅ ⟨z^{4}⟩ corresponds with A_{0}^{(5)} + B_{0}^{(5)}, which generates an 8-dimensional module

• ⟨1⟩ generates the entire module 𝔽D_{16}.

We now find all modules M and N such that 𝔽D_{16} ≅ M⊗N. Up to exchanging M and N, the modules must both be uniserial with dimensions 2 and 8. We give a table of all products bu, where b is one of the basis elements and u is a distinguished generator listed in section 4 of chapter 2. Where the product is relevant, we list the coefficient of A^{(1)} in the product.

A_{s,u,w,x}^{(8)} B_{s,u,w,x}^{(8)} A_{r}^{(2)} B^{(2)}
x̄(ȳx̄)^{3}  0  1  0  0
ȳ(x̄ȳ)^{3}  1  0  0  0
x̄  −  −  r  1
ȳ  −  −  1  0

Table of coefficients for A^{(1)} in (ID_{16})^{n}u

From this table we determine all products,

∥D_{16}∥ · m ⊗ n = x̄m ⊗ ȳ(x̄ȳ)^{3}n + ȳm ⊗ x̄(ȳx̄)^{3}n.

We list the coefficients of A^{(1)} ⊗ A^{(1)} in the product ∥D_{16}∥ · m ⊗ n in the following table.

A_{r}^{(2)} B^{(2)}
A_{s,u,w,x}^{(8)} r  1
B_{s,u,w,x}^{(8)} 1  0

Table of coefficients of A^{(1)} ⊗ A^{(1)} in ∥D_{16}∥ · m ⊗ n

From this table, we determine all tensor decompositions of 𝔽D_{16}:

Theorem 3.12. Up to isomorphism and rearrangement of factors, the following is the complete list of non-trivial tensor decompositions, 𝔽D_{16} ≅ M ⊗ N:

a) M ≅ 𝔽D_{16}A_{r}^{(2)} and N ≅ 𝔽D_{16}A_{s,u,w,x}^{(8)} when ρ ≠ 0;

b) M ≅ 𝔽D_{16}A_{r}^{(2)} and N ≅ 𝔽D_{16}B_{s,u,w,x}^{(8)}

c) M ≅ 𝔽D_{16}B^{(2)} and N ≅ 𝔽D_{16}A_{s,u,w,x}^{(8)}

The remaining group G is such that the Loewy length of 𝔽G is 6:

D_{8}Y4 = ⟨x, y, z | x^{4} = y^{2} = z^{2} = 1, x ∈ Z, (yz)^{2} = x^{2}⟩.

The expression x ∈ Z indicates that x is in the center of D_{8}Y4. This group is the central product of D_{8} and C_{4}.

The Poincaré polynomial associated with 𝔽D_{8}Y4 is P_{𝔽D_{8}Y4}(t) = 1+3t+4t^{2} + 4t^{3} + 3t^{4} + t^{5}. Assume 𝔽D_{8}Y4 ≅ M ⊗ N. If we assume dim(M) = dim(N) = 4 then 6 ≤ ℓℓ(M)+ℓℓ(N)−1 ≤ 4+4−1 = 7. At least one of the modules must have Loewy length 4, and is thus uniserial. But from Section 7 of Chapter 2 we see that there are no uniserial modules of length 4, so this is not a possibility. If we assume dim(M) = 2 then 6 ≤ 2 + ℓℓ(N) − 1 ≤ 1 + 5 = 6, as ℓℓ(N) < ℓℓ(𝔽D_{8}Y4). Thus we must have ℓℓ(N) = 5. The Poincaré polynomials for M and N are P_{M}(t) = 1 + t and P_{N}(t) = 1 + 2t + 2t^{2} + 2t^{3} + t^{4}.

Given M and N, we wish to determine if 𝔽G ≅ M ⊗ N. We only consider cases where dim(M) dim(N) = 16, ℓℓ(M) = 2 and ℓℓ(N) = 5. If we can show that the product ∥G∥ · M ⊗ N ≠ 0, then we know that 𝔽G ≅ M ⊗ N. Let m and n be generators of M and N respectively. By Lemma 1.11 of Chapter 1, the choice of generators does not matter. The product ∥G∥ · m ⊗ n is non-zero if and only if the product ∥G∥ · M ⊗ N is non-zero. We use Lemma 1.11 to find ∥G∥ · m ⊗ n. We find that

∥G∥ · (m ⊗ n) = x̄m ⊗ x̄^{2}ȳz̄n + ȳm ⊗ x̄^{3}z̄n + z̄m ⊗ x̄^{3}ȳn,

where m generates a module of length 2, and n generates a module of length 8, and where x̄, ȳ, and z̄ are defined in Section 7 of Chapter 2.

We first consider the decomposition of this group into the non-trivial product of subgroups H and K whose intersection is 1. We present the diagram for the full subgroup lattice of D_{8}Y4 after the theorem.

Theorem 3.13. The group ring 𝔽D_{8}Y4 is decomposable into the tensor product of permutation modules, 𝔽D_{8}Y4∥H∥ ⊗ 𝔽D_{8}Y4∥K∥. Up to interchanging H and K, these are the only such pairs of subgroups:

a) H is one of the four subgroups ⟨xy, xz⟩, ⟨x, y⟩, ⟨x, yz⟩, or ⟨xz, y⟩ and K is one of the conjugate subgroups ⟨z⟩ or ⟨x^{2}z⟩;

b) H is one of the four subgroups ⟨xy, xz⟩, ⟨x, z⟩, ⟨x, yz⟩, or ⟨xy, z⟩ and K is one of the conjugate subgroups ⟨y⟩ or ⟨x^{2}y⟩; c) H is one of the four subgroups ⟨xy, xz⟩, ⟨x, y⟩, ⟨x, z⟩, or ⟨y, z⟩ and K is one of the conjugate subgroups ⟨xyz⟩ or ⟨x^{−1}yz⟩;

Proof. Again, we observe the lattice diagram, and conclude that these are all the possible pairs of complementary subgroups in D_{8}Y4. Note that for each pair of statements of the theorem, there is an isomorphism on D_{8}Y4 which converts one of the statements to the other.

We now list, up to isomorphism, all permutation modules of 𝔽D_{8}Y4, identifying those that are uniserial on our list of all uniserial modules. We also identify 8-dimensional permutation modules that have Loewy length 5 with a generator described later. To every subgroup H of D_{8}Y4, there is a module 𝔽D_{8}Y4∥H∥. We list the subgroups in the order of decreasing size, with conjugate subgroups listed together. For each subgroup H, we list a generator of 𝔽D_{8}Y4∥H∥, as described in Chapter 2, section 7.

• ⟨x, y, z⟩ = D_{8}Y4 corresponds to A^{(1)}.

• ⟨xy, xz⟩ ≅ Q_{8} corresponds to A_{1,1}^{(2)}.

• ⟨x, y⟩ ≅ C_{2} × C_{4} corresponds to B_{0}^{(2)}.

• ⟨x, z⟩ ≅ C_{2} × C_{4} corresponds to C^{(2)}.

• ⟨x, yz⟩ ≅ C_{2} × C_{4} corresponds to B_{1}^{(2)}.

• ⟨z, xy⟩ ≅ D_{8} corresponds to A_{0,1}^{(2)}.

• ⟨y, xz⟩ ≅ D_{8} corresponds to A_{1,0}^{(2)}.

• ⟨y, z⟩ ≅ D_{8} corresponds to A_{0,0}^{(2)}.

• ⟨x⟩ ≅ C_{4} corresponds to x̄^{3}, which is not a uniserial module generator.

• ⟨xy⟩ ≅ C_{4} corresponds to x̄^{3} +x̄^{2}ȳ, which is not a uniserial module generator.

• ⟨xz⟩ ≅ C_{4} corresponds to x̄^{3} +x̄^{2}z̄, which is not a uniserial module generator.

• ⟨yz⟩ ≅ C_{4} corresponds to x̄^{2}(ȳ+z̄), which is not a uniserial module generator.

• ⟨x^{2}, y⟩ ≅ C_{2}^{2} corresponds to x̄^{2}ȳ, which is not a uniserial module generator.

• ⟨x^{2}, z⟩ ≅ C_{2}^{2} corresponds to x̄^{2}z̄, which is not a uniserial module generator.

• ⟨x^{2}, xyz⟩ ≅ C_{2}^{2} corresponds to x̄^{3} + x̄^{2}(ȳ + z̄ + ȳz̄), which is not a uniserial module generator.

• ⟨x^{2}⟩ ≅ C_{2} corresponds to x̄^{2}, which is not a uniserial module generator.

• C_{2} ≅ ⟨y⟩ and ⟨x^{2}y⟩ correspond to E_{0,0,0,0}^{(5)} and E_{1,0,0,0}^{(5)}.

• C_{2} ≅ ⟨z⟩ and ⟨x^{2}z⟩ correspond to F_{0,0,0,0}^{(5)} and F_{1,0,0,0}^{(5)}.

• C_{2} ≅ ⟨xyz⟩ and ⟨x^{−1}yz⟩ correspond to D_{1,0,0,0}^{(5)} and D_{0,0,0,0}^{(5)}.

• ⟨1⟩ generates all of 𝔽D_{8}Y4. We now find all modules M and N such that 𝔽D_{8}Y4 ≅ M ⊗ N. We have seen that if 𝔽D_{8}Y4 ≅ M ⊗ N then, up to exchanging M and N, the modules have dimensions 2 and 8, respectively, and N must have Loewy length 5, and P_{N} = 1 + 2t + 2t^{2} + 2t^{3} + t^{4}. We now find all cyclic submodules N of 𝔽D_{8}Y4 with Poincaré polynomial P_{N} = 1 + t + 2t^{2} + 2t^{3} + t^{4} + t^{5}. We use an algorithm similar to that described when we were dealing with G_{1}. We sketch the use of the algorithm, although the actual calculations are performed by computer, and are not shown here. We use the basis from Chapter 2, Section 7 for 𝔽D_{8}Y4. We start with an arbitrary element of Soc^{6}(𝔽D_{8}Y4),

D^{(5)} = \begin{aligned}& a \\ b & & c & & d \\ e & & f & g & & h \\ i & & j & k & & l \\ & m & & n & & p \\ & & & & q \end{aligned} .

Let N = 𝔽D_{8}Y4D^{(5)}. We need D^{(5)} ∉ Soc^{4}(𝔽D_{8}Y4). This implies that b, c and d can not all be zero. Consider the map from Rad(N) to Rad(N)/ Rad^{2}(N), where we list the elements in terms of the images of the basis elements x̄^{2}, x̄ȳ, x̄z̄, and ȳz̄ of 𝔽D_{8}Y4. Using the multiplications that are given in Chapter 2, Section 7, we find that the images of x̄D^{(5)}, ȳD^{(5)} and z̄D^{(5)} are (b, c, d, 0), (0, b, 0, d) and (c, 0, b, c) respectively. We may assume that the first non-zero element in the list {b, c, d} is equal to 1, as we may rescale. If b = 1, we must have cd = 1. If b = 0 and c = 1, then we must have d = 0. If b = c = 0, the case where d = 1 is also valid. Further quotients of powers of the radical show that all three of these outcomes are possible, with the added restriction that if b = 1 then c and d equal 1 as well.

For each of these three cases, we find the set of elements x̄^{i}ȳ^{j}z̄^{k}E^{(6)}, and perform a Gram-Schmidt type of process on the set in the same manner we described with G_{1} . In each of the three cases, we may assume that some of the entries are zero. In each case, there is also a restriction on the possible values of the remaining variables. We list the three possible families of distinguished module generators here, and rename them:

D_{r,s,t,u}^{(5)} = \begin{aligned}& · \\ 1 & & 1 & & 1 \\ · & & · & r & & 1 + r \\ · & & · & s & & t \\ & · & & · & & u \\ & & & & · \end{aligned} , E_{r,s,t,u}^{(5)} = \begin{aligned}& · \\ · & & 1 & & · \\ · & & · & · & & r \\ · & & · & s & & t \\ & · & & u & & · \\ & & & & · \end{aligned} ,

F_{r,s,t,u}^{(5)} = \begin{aligned}& · \\ · & & · & & 1 \\ r & & · & · & & · \\ s & & t & · & & · \\ & u & & · & & · \\ & & & & · \end{aligned}

We give a table of all products bu, where b is a one of the basis elements and u is a distinguished generator listed in Section 3 of Chapter 7, or one of those just described. We list the coefficient of A^{(1)} in the product.

D_{r,s,t,u}^{(5)} E_{r,s,t,u}^{(5)} F_{r,s,t,u}^{(5)}
x̄^{2}ȳz̄ 1  0  0
x̄^{3}z̄  1  1  0
x̄^{3}ȳ  1  0  1

Table of coefficients for A^{(1)} in (ID_{8}Y4)^{4}u

A_{r,s}^{(2)} B_{r}^{(2)} C^{(2)}
x̄ 1  0  0
ȳ s  r  1
z̄ r  1  0

Table of coefficients for A^{(1)} in (ID_{8}Y4)u From this table we determine all products,

∥D_{8}Y4∥ · m ⊗ n = x̄m ⊗ x̄^{2}ȳz̄n + ȳm ⊗ x̄^{3}z̄n + z̄m ⊗ x̄^{3}ȳn.

We list the coefficient of A^{(1)} ⊗A^{(1)} in the product ∥D_{8}Y4∥·m⊗n in the following table.

A_{ρ,σ}^{(2)} B_{ρ}^{(2)} C^{(2)}
D_{r,s,t,u}^{(5)} 1 + ρ + σ 1 + ρ 1
E_{r,s,t,u}^{(5)}  σ  ρ 1
F_{r,s,t,u}^{(5)}  ρ  1 0

Table of coefficients of A^{(1)} ⊗ A^{(1)} in ∥D_{8}Y4∥ · m ⊗ n From this table, we determine all tensor decompositions of 𝔽D_{8}Y4:

Theorem 3.14. Up to isomorphism and rearrangement of factors, the following is the complete list of non-trivial tensor decompositions, 𝔽D_{8}Y4 ≅ M ⊗ N:

a) M ≅ 𝔽D_{8}Y4A_{ρ,σ}^{(2)} and N ≅ 𝔽D_{8}Y4D_{r,s,t,u}^{(5)} when ρ + σ ≠ 1;

b) M ≅ 𝔽D_{8}Y4A_{ρ,σ}^{(2)} and N ≅ 𝔽D_{8}Y4E_{r,s,t,u}^{(5)} when σ ≠ 0;

c) M ≅ 𝔽D_{8}Y4A_{ρ,σ}^{(2)} and N ≅ 𝔽D_{8}Y4F_{r,s,t,u}^{(5)} when ρ ≠ 0;

d) M ≅ 𝔽D_{8}Y4B_{ρ}^{(2)} and N ≅ 𝔽D_{8}Y4D_{r,s,t,u}^{(5)} when ρ ≠ 1;

e) M ≅ 𝔽D_{8}Y4B_{ρ}^{(2)} and N ≅ 𝔽D_{8}Y4E_{r,s,t,u}^{(5)} when ρ ≠ 0;

f) M ≅ 𝔽D_{8}Y4B_{ρ}^{(2)} and N ≅ 𝔽D_{8}Y4F_{r,s,t,u}^{(5)};

g) M ≅ 𝔽D_{8}Y4C^{(2)} and N ≅ 𝔽D_{8}Y4D_{r,s,t,u}^{(5)};

h) M ≅ 𝔽D_{8}Y4C^{(2)} and N ≅ 𝔽D_{8}Y4E_{r,s,t,u}^{(5)}.

We may further factor P_{N}(t) = (1+t)(1+t+t^{2} +t^{3}), so it seems possible that 𝔽D_{8}Y4 may be factored into the tensor product of 3 modules, 𝔽D_{8}Y4 ≅ L⊗M ⊗N. We could rearrange the orders of L, M, and N so that dim(L) = dim(M) = 2 and dim(N) = 4. However, we know that 𝔽D_{8}Y4 has no decomposition into the product of two 4-dimensional modules. This implies that the modules L ⊗ M and N could not exist, so there is no non-trivial factorization of 𝔽D_{8}Y4 into a triple tensor product.

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Philip Lloyd Osterlund

178 words

Abstract

This thesis is aimed towards the determination of tensor decompositions of the regular representation of finite p-groups over fields of characteristic p. We also focus on uniserial modules, as they are often factors. We develop an algorithm that determines, up to isomorphism, all uniserial modules of a group ring (Section 2, pp. 11–12). In a tensor decomposition of the regular representation of a group, we relate the Loewy lengths, radical series and socle series of the regular representation to those of the factors (Theorems 1.4–1.7, pp. 69–74). Under certain conditions, we find that the Poincaré polynomial associated to the filtration of a module is the product of the corresponding Poincaré polynomials of the module’s tensor factors (Corollary 1.8, p. 73). For the non-abelian groups of order 16 that can not be written as direct products of proper subgroups we classify the uniserial modules (Chapter 2, pp. 23–66) and all tensor decompositions (Section 3, pp. 85–111).

Key words Tensor decomposition, uniserial module, p-group, radical and socle series, Poincaré polynomial.

Contents

Chapter 0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 1 Classifying Uniserial Modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 §1 Uniserial Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 §2 A Procedure for Finding Uniserial Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 §3 An Example of the Procedure for Finding Uniserial Submodules of 𝔽G . . 13 Chapter 2 Classification of Uniserial Modules for Groups of Order 16 . . . . . . . . . 21 §1 Uniserial submodules of 𝔽G_{1} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 §2 Uniserial submodules of 𝔽(2 × 4).2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31 §3 Uniserial submodules of 𝔽Mod_{16} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36 §4 Uniserial submodules of 𝔽D_{16} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 §5 Uniserial submodules of 𝔽SD_{16} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 §6 Uniserial submodules of 𝔽Q_{16} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 §7 Uniserial submodules of 𝔽D_{8}Y4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64 Chapter 3 Tensor decompositions of the Regular Representation . . . . . . . . . . . . . . 67 §1 Tensor Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 §2 Permutation Module Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 §3 Tensor Decompositions of 𝔽G, where |G| = 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Figures

Uniserial Submodule Poset for 𝔽G_{1} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Uniserial Module Homomorphism Poset for 𝔽G_{1} . . . . . . . . . . . . . . . . . . . . . . . . . 23

Uniserial Submodule Poset for 𝔽(2 × 4).2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Uniserial Module Homomorphism Poset for 𝔽(2 × 4).2 . . . . . . . . . . . . . . . . . . . 31

Uniserial Submodule Poset for 𝔽Mod_{16} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Uniserial Module Homomorphism Poset for 𝔽Mod_{16} . . . . . . . . . . . . . . . . . . . . . 37

Uniserial Submodule Poset for 𝔽D_{16} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Uniserial Module Homomorphism Poset for 𝔽D_{16} . . . . . . . . . . . . . . . . . . . . . . . . 44

Uniserial Submodule Poset for 𝔽SD_{16} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Uniserial Module Homomorphism Poset for 𝔽SD_{16} . . . . . . . . . . . . . . . . . . . . . . .51

Uniserial Submodule Poset for 𝔽Q_{16} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Uniserial Module Homomorphism Poset for 𝔽Q_{16} . . . . . . . . . . . . . . . . . . . . . . . . 58

Uniserial Submodule Poset for 𝔽D_{8}Y4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Uniserial Module Homomorphism Poset for 𝔽D_{8}Y4 . . . . . . . . . . . . . . . . . . . . . . 64

Subgroup lattice of G_{1} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Subgroup lattice of (2 × 4).2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93

Subgroup lattice of Mod_{16} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Subgroup lattice of SD_{16} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Subgroup lattice of Q_{16} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Subgroup lattice of D_{16} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Subgroup lattice of D_{8}Y4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Tables

Table of elements of (IG_{1})u and (IG_{1})^{5}u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Table of coefficients for A^{(1)} ⊗ A^{(1)} in ∥G_{1}∥ · m ⊗ n . . . . . . . . . . . . . . . . . . . . . . 91

Table of elements of (IG_{1})^{3} u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Table of coefficients of A^{(1)} ⊗ A^{(1)} in ∥G∥ · m ⊗ n . . . . . . . . . . . . . . . . . . . . . . . . 91

Table of elements of (I(2 × 4).2)^{3} u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Table of coefficients of A^{(1)} ⊗ A^{(1)} in ∥(2 × 4).2∥ · m ⊗ n . . . . . . . . . . . . . . . . . 95

Table of coefficients for A^{(1)} in (IMod_{16})^{n}u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Table of coefficients of A^{(1)} ⊗ A^{(1)} in ∥Mod_{16}∥ · m ⊗ n . . . . . . . . . . . . . . . . . . . 98

Table of coefficients for A^{(1)} in (ISD_{16})^{n}u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Table of coefficients of A^{(1)} ⊗ A^{(1)} in ∥SD_{16}∥ · m⊗n . . . . . . . . . . . . . . . . . . . . . 101

Table of coefficients for A^{(1)} in (ID_{16})^{n}u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Table of coefficients of A^{(1)} ⊗ A^{(1)} in ∥D_{16}∥ · m ⊗ n . . . . . . . . . . . . . . . . . . . . . 105

Table of coefficients for A^{(1)} in (ID_{8}Y4)^{4}u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Table of coefficients for A^{(1)} in (ID_{8}Y4)u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Table of coefficients of A^{(1)} ⊗ A^{(1)} in ∥D_{8}Y4∥ · m ⊗ n . . . . . . . . . . . . . . . . . . . 110