The vertices and sources of the natural simple module for the alternating group in even characteristic

Susanne Danz and Jürgen Müller

Abstract

For n ≥ 5 the natural permutation module for the alternating group 𝔄_{n} has a unique non-trivial composition factor, being called its natural simple module. We determine the vertices and sources of the natural simple 𝔄_{n}-module over fields of characteristic 2. Mathematics Subject Classification: 20C20, 20C30

1 Introduction and result

One of the leading themes in the representation theory of finite groups is the question how far the p-modular representation theory of a given group is determined by local data, that is by its p-subgroups and their normalisers. From this viewpoint, it is immediate to ask for the vertices and sources of the building blocks of all modules, namely the simple ones. Although not too much seems to be known in general, in recent years the picture has changed considerably for the symmetric group 𝔖_{n} on n letters, see for example the survey [3]. Now it is natural to ask how the vertices and sources of the simple 𝔖_{n}-modules and those of the constituents of their restrictions to the alternating group 𝔄_{n} are related, and here in particular the natural 𝔖_{n}-module immediately springs to mind. The vertices and sources of the natural 𝔖_{n}-module have been determined in [13], and indeed the latter paper is the starting point of the present work. As it turns out, the case of even characteristic is by far the most interesting one.

Hence, let F be an algebraically closed field of characteristic 2, let n ≥ 3, and let D := D^{(n−1,1)} be the natural simple F 𝔖_{n}-module, that is, D is the unique non-trivial composition factor of the natural permutation F 𝔖_{n}-module M := M^{(n−1,1)} := Ind_{𝔖_{n−1}}^{𝔖_{n}}(F). Now Res_{𝔄_{n}}^{𝔖_{n}}(D)

splits into two non-isomorphic simple modules E_{±}^{(n−1,1)} if n < 5, and otherwise Res_{𝔄_{n}}^{𝔖_{n}}(D) =: E_{0}^{(n−1,1)} remains simple (cf. [1]). In consequence of Mackey’s Decomposition Theorem, we deduce that E_{0}^{(n−1,1)} is the unique non-trivial composition factor of the natural permutation F 𝔄_{n}-module Ind_{𝔄_{n−1}}^{𝔄_{n}}(F) ≅ Res_{𝔄_{n}}^{𝔖_{n}}(M) if n ≥ 5, and we call E_{0}^{(n−1,1)} the natural simple F 𝔄_{n}-

module. If n < 5 then Ind_{𝔄_{n−1}}^{𝔄_{n}}(F) has two non-trivial composition factors, E_{±}^{(2,1)} and E_{±}^{(3,1)}, respectively, also called the natural simple F 𝔄_{3}- and F 𝔄_{4}-modules, respectively.

Our main aim now is to prove the following theorem. Part of the assertions, listed for the sake of completeness, are fairly immediate, and already covered in the subsequent remark. There we also comment on the odd characteristic case, and recall the results on the vertices and sources of the module D from [13].

Theorem 1.1. Let F be an algebraically closed field of characteristic 2, and n ≥ 3. Let E be the natural simple F 𝔄_{n}-module, and let further Q be a vertex of E. Then the following hold:

(i) If n is odd then Q is conjugate to a Sylow 2-subgroup of 𝔄_{n−3}, and E has trivial source.

(ii) If n > 6 is even then Q is a Sylow 2-subgroup of 𝔄_{n}, and Res_{Q}^{𝔄_{n}}(E) is a source of E.

(iii) If n = 4 then Q is the Sylow 2-subgroup of 𝔄_{4}, and E has trivial source. If n = 6 then Q is conjugate to (𝔖_{2} × 𝔖_{2} × 𝔖_{2}) ∩ 𝔄_{6}, and E has a source of dimension 2.

Remark 1.2. (a) For the moment, we consider representations over an algebraically closed field of odd characteristic p, and let p ≤ n. Then the natural simple module D is not isomorphic to its conjugate counterpart D′ := D ⊗ sgn, where sgn denotes the sign representation. Hence we have the natural simple F 𝔄_{n}-module E := Res_{𝔄_{n}}^{𝔖_{n}}(D) ≅ Res_{𝔄_{n}}^{𝔖_{n}}(D′). By [13, Thm. 1.2, 1.3], the vertices of D coincide with the defect groups of its block, which are contained in 𝔄_{n} anyway and thus are the vertices of E as well. Moreover, D and E have the same sources, which are trivial if p ∤ n, while if p | n then are the restrictions of D to its vertices. This settles the case of odd characteristic, hence from now on we again stick to even characteristic.

(b) In the case n = 3, the simple F 𝔖_{3}-module D^{(2,1)} as well as the simple F 𝔄_{3}-modules E_{±}^{(2,1)} are projective and have thus trivial sources. If n > 3 is odd then D^{(n−1,1)} ≅ S^{(n−1,1)}, where S^{(n−1,1)} is the natural Specht F 𝔖_{n}-module, and E := E_{0}^{(n−1,1)} = Res_{𝔄_{n}}^{𝔖_{n}}(D) is simple. By [13, Thm. 1.2, 1.3], the vertices of D are precisely the defect groups of its block, which are conjugate to a Sylow 2-subgroup of 𝔖_{n−3}, and D has trivial source. Let Q_{n−3} be a Sylow 2-subgroup of 𝔄_{n−3}. This is a defect group of the block of F 𝔄_{n} containing E. Since D has trivial source, we also deduce that F is a direct summand of Res_{Q_{n−3}}^{𝔄_{n}}(E). Therefore, also E has the defect groups of its block as vertices and trivial source.

(c) If n is even then D is contained in the principal block of F 𝔖_{n}. By [13, Thm. 1.4(b), 1.5(b)], D^{(3,1)} has the Sylow 2-subgroup of 𝔄_{4} as vertex and trivial source.

In [13, Thm. 1.4(a), 1.5(a)] it is stated that D^{(n−1,1)} has the Sylow 2-subgroup of 𝔖_{n} as vertex, and its restriction to the vertex as source, for n ≥ 6. This statement is correct, alone the proof of [13, Prop. 4.5] used in between has a gap. It occurs in the very last line of [13], and we have detected it while writing the present paper. Actually, we are able to use the techniques developed here to close that gap, and to give a new complete proof of [13, Thm. 1.4(a), 1.5(a)]. Thus, in the present paper we only use [13, Thm. 1.2, 1.3, 1.4(b), 1.5(b)] as well as [13, L. 3.3, Prop. 4.2], which all are valid with the proofs given, at least as far as we see, and we have made sure that we do not refer to any results which have been proven elsewhere using [13].

Remark 1.3. Now the task is to determine the vertices of the natural simple F 𝔄_{n}-module E := E_{0}^{(n−1,1)} whenever n ≥ 6 is even. Hence from now on we suppose that n ≥ 4 is even. Suppose further that Q_{n} is a Sylow 2-subgroup of 𝔄_{n}, and that Q ≤ Q_{n} is a vertex of E.

(a) Then our general strategy is to show, firstly, that Ẽ := Res_{Q_{n}}^{𝔄_{n}}(E) is indecomposable. This implies that Q ≤ Q_{n} also is a vertex of Ẽ. Then, secondly, we assume that Ẽ is relatively projective with respect to some maximal subgroup R < Q_{n} containing Q. In particular, we have Φ(Q_{n}) ≤ R, where Φ(Q_{n}) denotes the Frattini subgroup of Q_{n}. Moreover, we have Res_{𝔄_{n−1}}^{𝔄_{n}}(E) ≅ E_{0}^{(n−2,1)}, where, by Remark 1.2, the latter has vertex Q_{n−4} and trivial source T. Thus T is a direct summand of Res_{Q_{n−4}}^{Q_{n}}(Ẽ), hence letting S be a source of Ẽ we infer that T is a direct summand of Res_{Q_{n−4}}^{Q_{n}}(Ind_{Q}^{Q_{n}}(S)). Hence Mackey’s Decomposition Theorem implies that there is g ∈ Q_{n} such that Q^{g} ∩ Q_{n−4} = Q_{n−4} (cf. [14, L. 4.3.4]), that is Q_{n−4} ≤_{Q_{n}} Q ≤ R < Q_{n}. Thus, since R is normal in Q_{n}, we have Q_{n−4} ≤ R as well. In summary we have ⟨Φ(Q_{n}), Q_{n−4}, Q⟩ ≤ R, which typically turns out to be a fairly restrictive condition on R.

Now, Ẽ being relatively projective with respect to R, Green’s Indecomposability Theorem implies that Ẽ is induced from an indecomposable direct summand of Res_{R}^{Q_{n}}(Ẽ), implying that Res_{R}^{Q_{n}}(Ẽ) is the direct sum of two indecomposable modules of dimension dim(E)/2 each. This latter conclusion is then drawn to a contradiction, implying that Ẽ, and thus E, cannot possibly be relatively projective with respect to any proper subgroup of Q_{n}.

(b) In the sequel, a key player will be the natural permutation F 𝔖_{n}-module M. Letting {γ_{1}, …, γ_{n}} be its permutation basis, M is uniserial with composition series M > M′ > M′′ > 0, where

M′ := {∑_{i=1}^{n} a_{i}γ_{i} ∈ M | a_{1}, …, a_{n} ∈ F, ∑_{i=1}^{n} a_{i} = 0},

M′′ := {a ∑_{i=1}^{n} γ_{i} ∈ M | a ∈ F } ≅ F ≅ M/M′,

and M′/M′′ ≅ D as F 𝔖_{n}-modules. In particular, dim(M′) = n − 1 and dim(D) = n − 2. Moreover, {γ_{i} + γ_{n} | i ∈ {1, …, n − 1}} is an F -basis of M′, and if ^{–} : M ⟶ M/M′′ denotes the natural epimorphism then \{\overline{γ}_{i} + \overline{γ}_{n} | i ∈ \{1, …, n − 2\}\} is an F -basis of M′/M′′ (cf. [9, Ex. 5.1]). For convenience, in the following we will simply identify D with M′/M′′.

This paper now is organised as follows: In Section 2 we begin by collecting some properties of the Sylow 2-subgroups of 𝔖_{n} and 𝔄_{n}, and their subgroups. It turns out that the behaviour of E depends on the 2-adic expansion of n. Hence letting n = ∑_{j=1}^{l} 2^{i_{j}}, for some l ≥ 1 and i_{1} > … > i_{l} ≥ 1, we distinguish between several cases. In Section 3 we settle the case l = 1, that is, n is a 2-power. In Section 4 we begin to investigate the case l ≥ 2, that is, n is even but not a 2-power, by first considering restrictions of E to various abelian subgroups. This leads to further case distinctions with respect to n_{l} := 2^{i_{l}}. In Section 5 we settle the case n_{l} > 2, while for n_{l} = 2 in Sections 6–8 the cases l ≥ 4, l = 3, and l = 2, are dealt with, respectively. In the very end the case n = 6 remains to be considered, completing the proof of Theorem 1.1. In the final Section 9, which is an appendix, we in particular use the technique employed in Section 8 to give the new proof of [13, Thm. 1.4(a), 1.5(a)].

We want to point out that, although no explicit reference to computer calculations is made in the present paper, before writing it we have dealt with various examples by way of explicit computations, which helped us to understand the different behaviour of E with respect to the 2-adic expansion of n. To do so, we have used the computer algebra systems GAP [6] and MAGMA [2], and the specially tailored computational techniques to determine vertices and sources developed in [4]. Finally, for any finite group G, an F G-module is understood to be a finitely generated right F G-module. Furthermore, the endomorphism algebra End_{F G}(M) is also supposed to act from the right. For an introduction to the theory of vertices and sources we refer to [14, Sec. 4.3], and for details concerning the representation theory of the symmetric groups to [9] and [10].

Acknowledgement. The first author’s research on this article has been supported by the Deutsche Forschungsgemeinschaft through DFG grant #DA 1115/1-1. The second author gratefully acknowledges financial support and the enjoyable hospitality of Friedrich-Schiller-Universität Jena, where parts of the present paper have been written.

2 Sylow subgroups of the symmetric and alternating groups

We begin by introducing our notation for the Sylow 2-subgroups of 𝔖_{n} and 𝔄_{n}, respectively. Moreover, we collect some properties of these Sylow subgroups and their subgroups which will be needed in the course of the subsequent sections.

Remark 2.1. (a) First of all, let n = 2^{m}, for some m ∈ ℕ_{0}. We set P_{1} := 1, P_{2} := C_{2} := ⟨(1, 2)⟩, and P_{2^{i+1}} := P_{2^{i}} ≀ C_{2} = {(x_{1}, x_{2}; σ) | x_{1}, x_{2} ∈ P_{2^{i}}, σ ∈ P_{2}}, for i ≥ 1. Recall that, for i ≥ 1 and (x_{1}, x_{2}; σ), (y_{1}, y_{2}; π) ∈ P_{2^{i+1}}, we have (x_{1}, x_{2}; σ)(y_{1}, y_{2}; π) = (x_{1}y_{1^{σ}}, x_{2}y_{2^{σ}}; σπ). As usual, for i ≥ 0, we regard P_{2^{i}} as a subgroup of 𝔖_{2^{i}} in the obvious way. Then, by [10, 4.1.22, 4.1.24], P_{n} = P_{2^{m}} is a Sylow 2-subgroup of 𝔖_{2^{m}} = 𝔖_{n}.

Moreover, by construction, P_{2^{m}} is generated by the following elements:

w_{2^{i}} := ∏_{k=1}^{2^{i−1}}(k, k + 2^{i−1}),  (1)

for i = 1, …, m. For instance, P_{8} is generated by w_{2} = (1, 2), w_{4} = (1, 3)(2, 4) and w_{8} = (1, 5)(2, 6)(3, 7)(4, 8). Additionally let w_{1} := 1. Since, by [8, Satz III.15.3], we have |P_{2^{m}} : Φ(P_{2^{m}})| = 2^{m}, (1) yields in fact a minimal set of generators for P_{n} = P_{2^{m}}.

If m ≥ 2, then we have P_{n} = P_{2^{m}} = P_{2^{m−1}} ≀ P_{2} = {(x_{1}, x_{2}; σ) | x_{1}, x_{2} ∈ P_{2^{m−1}}, σ ∈ P_{2}}. The base group of P_{2^{m−1}} ≀P_{2} shall henceforth be denoted by B_{n}. When viewing P_{2^{m}} as a subgroup of 𝔖_{2^{m}} as above, B_{n} corresponds to P_{2^{m−1}} × w_{n}P_{2^{m−1}}w_{n}.

(b) Let now n ≥ 2 be even with 2-adic expansion n = ∑_{j=1}^{l} 2^{i_{j}}, for appropriate l ≥ 2 and i_{1} > … > i_{l} ≥ 1, and let n_{j} := 2^{i_{j}} for j = 1, …, l. Then, by [10, 4.1.22, 4.1.24], P_{n} = ∏_{j=1}^{l} P_{n_{j}}^{g_{j}}, with g_{1} = 1, and g_{j} := ∏_{k=1}^{n_{j}}(k, k + ∑_{s=1}^{j−1} n_{s}) for j = 2, …, l, is a Sylow 2subgroup of 𝔖_{n}. For convenience, we will simply write P_{n} = ∏_{j=1}^{l} P_{n_{j}} where P_{n_{j}} is understood to be acting on the subset Ω_{j} := {1 + ∑_{s=1}^{j−1} n_{s}, …, ∑_{s=1}^{j} n_{s}} of {1, …, n}, for j = 1, …, l. If n is odd then we simply set P_{n} := P_{n−1}, so that again P_{n} is a Sylow 2-subgroup of 𝔖_{n}.

Via (1) we obtain a minimal set of generators for P_{n_{j}} denoted by W_{j} := {w_{2,j}, …, w_{n_{j},j}}, for j = 1, …, l. That is, W := ⋃_{j=1}^{l} W_{j} is a minimal generating set for P_{n}. For instance, P_{14} = P_{8} × P_{4} × P_{2} is generated by w_{2,1} = (1, 2), w_{4,1} = (1, 3)(2, 4), w_{8,1} = (1, 5)(2, 6)(3, 7)(4, 8), w_{2,2} = (9, 10), w_{4,2} = (9, 11)(10, 12) and w_{2,3} = (13, 14).

(c) Now, for any n ≥ 2, we set Q_{n} := P_{n} ∩ 𝔄_{n} so that Q_{n} is a Sylow 2-subgroup of 𝔄_{n}.

If n = 2^{m}, for some m ≥ 2, then writing P_{n} = P_{2^{m−1}} ≀ P_{2} we have Q_{n} = {(x_{1}, x_{2}; σ) ∈ P_{2^{m−1}} ≀ P_{2} | x_{1}x_{2} ∈ Q_{2^{m−1}}} ≤ P_{2^{m−1}} ≀ P_{2}, and we define

w′_{2} := w_{2} · w_{2}^{w_{2^{m}}} = (1, 2)(2^{m−1} + 1, 2^{m−1} + 2), and w′_{2^{i}} := w_{2^{i}} for i = 2, …, m. (2)

If n = ∑_{j=1}^{l} 2^{i_{j}} = ∑_{j=1}^{l} n_{j}, for some l ≥ 2 and some i_{1} > … > i_{l} ≥ 1, is the 2-adic expansion of n as above then we define

w′_{2,j} := w_{2,j}w_{2,l} for j = 1, …, l − 1, and w′_{2^{s},j} := w_{2^{s},j} for j = 1, …, l and s = 2, …, i_{j}. (3)

Proposition 2.2. With the notation of the previous remark we have:

(i) If n = 2^{m}, for some m ≥ 2, then Q_{n} = ⟨w′_{2}, …, w′_{2^{m}}⟩. Furthermore, {w′_{2}, …, w′_{2^{m}}} is a minimal set of generators for Q_{n}.

(ii) If n = ∑_{j=1}^{l} 2^{i_{j}} = ∑_{j=1}^{l} n_{j}, for some l ≥ 2 and some i_{1} > … > i_{l} ≥ 1, then

W′ := ⋃_{j=1}^{l−1}{w′_{2,j}, …, w′_{n_{j},j}} ∪ {w′_{4,l}, …, w′_{n_{l},l}}

is a minimal generating set for Q_{n}.

Proof. Suppose first that n = 2^{m}, for some m ≥ 2. Obviously, Q := ⟨w′_{2}, …, w′_{2^{m}}⟩ ≤ Q_{n}. Moreover, we have w_{2}w′_{2^{i}}w_{2} = w′_{2}w′_{2^{i}}w′_{2} ∈ Q, for i = 1, …, m − 1, and

w_{2}w′_{2^{m}}w_{2} = w_{2}w_{2^{m}}w_{2} · w_{2^{m}}w_{2^{m}} = w′_{2}w′_{2^{m}} ∈ Q,

so that w_{2}Qw_{2} = Q. This shows that ⟨w_{2}⟩Q = Q⟨w_{2}⟩ = P_{n}, and thus |P_{n} : Q| = 2, that is Q = Q_{n}. It remains to show that {w′_{2}, …, w′_{2^{m}}} is a minimal generating set for Q_{n}. For this, notice that Q_{n} acts transitively on {1, …, n}, whereas for any j ∈ {1, …, m} the group Q := ⟨w′_{2^{i}} | i ∈ {1, …, m}\{j}⟩ does not, namely, 1 and 1+2^{j−1} then lie in different Q-orbits. Consequently, {w′_{2}, …, w′_{2^{m}}} is in fact minimal.

Now suppose that n = ∑_{j=1}^{l} 2^{i_{j}} = ∑_{j=1}^{l} n_{j}, for some l ≥ 2 and some i_{1} > … > i_{l} ≥ 1. Then Q := ⟨W′⟩ ≤ Q_{n}. Furthermore, we have

w_{2,l}w′_{2^{s},j}w_{2,l} = w′_{2^{s},j} ∈ Q, for j = 1, …, l − 1, s = 1, …, i_{j},
w_{2,l}w′_{2^{s},l}w_{2,l} = w′_{2,1}w′_{2^{s},l}w′_{2,1} ∈ Q, for s = 2, …, i_{l},

hence w_{2,l}Qw_{2,l} = Q, and clearly ⟨w_{2,l}⟩Q = Q⟨w_{2,l}⟩ = P_{n}. This shows that |P_{n} : Q| = 2 so that Q = Q_{n}. Furthermore, from Remark 2.1 we deduce that W′ ∪ {w_{2,l}} is a minimal generating set for P_{n}, and thus also W′ has to be a minimal generating set for Q_{n}.

Example 2.3. In order to illustrate the rather technical notation above, consider the cases where n = 8 and n = 14 = 8 + 4 + 2, respectively: Then Q_{8} is generated by

w′_{2} = (1, 2)(5, 6), w′_{4} = (1, 3)(2, 4), w′_{8} = (1, 5)(2, 6)(3, 7)(4, 8).

Analogously, Q_{14} is generated by

w′_{2,1} = (1, 2)(13, 14),  w′_{4,1} = (1, 3)(2, 4),  w′_{8,1} = (1, 5)(2, 6)(3, 7)(4, 8),
w′_{2,2} = (9, 10)(13, 14),  w′_{4,2} = (9, 11)(10, 12).

Proposition 2.4. For any n ≥ 2 we have Φ(P_{n}) = [P_{n}, P_{n}] and Φ(Q_{n}) = [Q_{n}, Q_{n}]. If n = 2^{m}, for some m ≥ 2, then |Φ(P_{n}) : Φ(Q_{n})| = 2, otherwise Φ(P_{n}) = Φ(Q_{n}).

Proof. By [8, Satz III.3.14], [P_{n}, P_{n}] ≤ Φ(P_{n}) and [Q_{n}, Q_{n}] ≤ Φ(Q_{n}). As we have just seen, both P_{n} and Q_{n} are generated by elements of order 2. Consequently, the same holds true for P_{n}/[P_{n}, P_{n}] and Q_{n}/[Q_{n}, Q_{n}]. Thus both P_{n}/[P_{n}, P_{n}] and Q_{n}/[Q_{n}, Q_{n}] are elementary abelian. This yields [P_{n}, P_{n}] = Φ(P_{n}) and [Q_{n}, Q_{n}] = Φ(Q_{n}). The second assertion is immediate from Proposition 2.2 and Remark 2.1.

Proposition 2.5. Let n = 2^{m}, for some m ≥ 2. Then we have Φ(P_{n}) = {(x_{1}, x_{2}; 1) | x_{1}x_{2} ∈ Φ(P_{2^{m−1}})} and Φ(Q_{n}) = {(x_{1}, x_{2}; 1) | x_{1}, x_{2} ∈ Q_{2^{m−1}}, x_{1}x_{2} ∈ Φ(P_{2^{m−1}})}.

Proof. The first assertion follows from [15, L. 1.4] and the fact that Φ(P_{n}) = [P_{n}, P_{n}]. It remains to show that Φ(Q_{n}) = H := {(x_{1}, x_{2}; 1) | x_{1}, x_{2} ∈ Q_{2^{m−1}}, x_{1}x_{2} ∈ Φ(P_{2^{m−1}})} ≤ Φ(P_{n}). For this let x = (x_{1}, x_{2}; σ) ∈ Q_{n}. Then x_{1}, x_{2} ∈ P_{2^{m−1}}, x_{1}x_{2}, x_{2}x_{1} ∈ Q_{2^{m−1}} and σ ∈ {(1, 2), 1}. Moreover,

x^{2} = \begin{cases}(x_{1}^{2}, x_{2}^{2}; 1), & if σ = 1, \\ (x_{1}x_{2}, x_{2}x_{1}; 1), & if σ ≠ 1. \end{cases}

That is, x^{2} = (y_{1}, y_{2}; 1) where y_{1}, y_{2} ∈ Q_{2^{m−1}}, and x^{2} ∈ Φ(P_{n}), by [8, Satz III.3.14]. Thus y_{1}y_{2} ∈ Φ(P_{2^{m−1}}). Since, again by [8, Satz III.3.14], Φ(Q_{n}) is generated by the squares in Q_{n}, this implies Φ(Q_{n}) ≤ H ≤ Φ(P_{n}). But Q_{2^{m−1}} < P_{2^{m−1}} so that H ≠ Φ(P_{n}). Therefore, from Proposition 2.4 we deduce that H = Φ(Q_{n}).

Proposition 2.6. Suppose that n = 2^{m}, for some m ≥ 2. Then the following hold:

(i) We have B_{n} = ⟨w_{2}, …, w_{2^{m−1}}, w_{2}^{w_{n}}, …, w_{2^{m−1}}^{w_{n}}⟩ and

B_{n} ∩ Q_{n} = ⟨w_{2}w_{2}^{w_{n}}, w_{4}, …, w_{2^{m−1}}, w_{4}^{w_{n}}, …, w_{2^{m−1}}^{w_{n}}⟩.

(ii) We have P_{2^{m−1}}Φ(P_{n}) = B_{n} and Q_{2^{m−1}}Φ(P_{n}) = B_{n} ∩ Q_{n}.

(iii) If m ≥ 3 then we have Q_{2^{m−1}+2}Φ(Q_{n}) = B_{n} ∩ Q_{n}.

Proof. Since B_{n} = P_{2^{m−1}} × P_{2^{m−1}}^{w_{n}}, the first statement in (i) is clear from Remark 2.1. Furthermore, ⟨w_{2}w_{2}^{w_{n}}, w_{4}, …, w_{2^{m−1}}, w_{4}^{w_{n}}, …, w_{2^{m−1}}^{w_{n}}⟩ ≤ B_{n} ∩ 𝔄_{n} = B_{n} ∩ Q_{n}, and is normalized by w_{2}. Since ⟨w_{2}⟩ · ⟨w_{2}w_{2}^{w_{n}}, w_{4}, …, w_{2^{m−1}}, w_{4}^{w_{n}}, …, w_{2^{m−1}}^{w_{n}}⟩ = B_{n}, also the second assertion in (i) follows.

Now we show that B_{n} ≤ P_{2^{m−1}}Φ(P_{n}). For this let x ∈ B_{n}, and write x = (x_{1}, x_{2}; 1) with x_{1}, x_{2} ∈ P_{2^{m−1}}. Then x = (x_{1}x_{2}, 1; 1)(x_{2}^{−1}, x_{2}; 1). Since x_{1}x_{2} ∈ P_{2^{m−1}} and (x_{2}^{−1}, x_{2}; 1) ∈ Φ(P_{n}), we get x ∈ P_{2^{m−1}}Φ(P_{n}). Consequently, P_{2^{m−1}}Φ(P_{n}) is a proper subgroup of P_{n} containing the maximal subgroup B_{n} of P_{n}, and hence B_{n} = P_{2^{m−1}}Φ(P_{n}).

Next we verify that B_{n} ∩ Q_{n} = Q_{2^{m−1}}Φ(P_{n}). By (i), we already know that Q_{2^{m−1}}Φ(P_{n}) ≤ (P_{2^{m−1}}Φ(P_{n})) ∩ Q_{n} = B_{n} ∩ Q_{n}. Now let x ∈ B_{n} ∩ Q_{n}, and again write x = (x_{1}, x_{2}; 1) for appropriate x_{1}, x_{2} ∈ P_{2^{m−1}}. Since x ∈ Q_{n}, we have x_{1}x_{2} ∈ Q_{2^{m−1}}, and thus x = (x_{1}x_{2}, 1; 1)(x_{2}^{−1}, x_{2}; 1) ∈ Q_{2^{m−1}}Φ(P_{n}). Hence Q_{2^{m−1}}Φ(P_{n}) = B_{n} ∩ Q_{n}, proving (ii).

Now let m ≥ 3, then Q_{2^{m−1}+2}Φ(Q_{n}) ≤ (P_{2^{m−1}+2}Φ(P_{n})) ∩ Q_{n} = B_{n} ∩ Q_{n}. The last equation follows from the fact that P_{2^{m−1}+2} < P_{n} so that P_{2^{m−1}+2}Φ(P_{n}) is a proper subgroup of P_{n} containing the maximal subgroup B_{n} = P_{2^{m−1}}Φ(P_{n}). Since

Φ(Q_{n}) ∩ Q_{2^{m−1}+2} = {(x_{1}, 1; 1) | x_{1} ∈ Φ(P_{2^{m−1}})} = Q_{2^{m−1}} ∩ Φ(P_{n}),

we now have

|Q_{2^{m−1}+2}Φ(Q_{n})| = \frac{|Q_{2^{m−1}+2}||Φ(Q_{n})|}{|Φ(Q_{n}) ∩ Q_{2^{m−1}+2}|} = \frac{|Q_{2^{m−1}+2}||Φ(Q_{n})|}{|Φ(P_{2^{m−1}})|} = \frac{|Q_{2^{m−1}}| · 2 · |Φ(Q_{n})|}{|Φ(P_{2^{m−1}})|}

= \frac{|Q_{2^{m−1}}||Φ(P_{n})|}{|Φ(P_{2^{m−1}})|} = |Q_{2^{m−1}}Φ(P_{n})| = |B_{n} ∩ Q_{n}|.

This finally shows that also Q_{2^{m−1}+2}Φ(Q_{n}) = B_{n} ∩ Q_{n}, and assertion (iii) follows.

Remark 2.7. (a) Again let n = ∑_{j=1}^{l} 2^{i_{j}} = ∑_{j=1}^{l} n_{j}, for some l ≥ 1 and some i_{1} > … > i_{l} ≥ 1, be the 2-adic expansion of n. For j = 1, …, l define y_{n_{j}} := w_{n_{j},j} ⋯ w_{2,j}, where the w_{2,j}, …, w_{n_{j},j} are as in Remark 2.1. Then, by [13, L. 3.3], y_{n_{j}} is an n_{j}-cycle, for j = 1, …, l. Furthermore, let Y_{n_{j}} := ⟨y_{n_{j}}⟩, for j = 1, …, l, and Y_{n} := ∏_{j=1}^{l} Y_{n_{j}}; clearly Y_{n} ≰ 𝔄_{n}. Next we set y′_{n_{j}} := y_{n_{j}}y_{n_{l}} and Y′_{n_{j}} := ⟨y′_{n_{j}}⟩. Then we have Y′_{n} := Y_{n} ∩ 𝔄_{n} = ∏_{j=1}^{l} Y′_{n_{j}}, namely, we clearly have Y := ⟨y′_{n_{1}}, …, y′_{n_{l}}⟩ ≤ Y′_{n}, and from ⟨y_{n_{l}}⟩Y = Y ⟨y_{n_{l}}⟩ = Y_{n} where y_{n_{l}} ∉ Y but y_{n_{l}}^{2} ∈ Y we get |Y_{n} : Y | = 2, hence Y = Y′_{n}.

Moreover, we set x_{n_{j}} := y_{n_{j}}^{2}, for j = 1, …, l, that is, x_{n_{j}} has cycle type (n_{j}/2, n_{j}/2), for j = 1, …, l. We also set X_{n_{j}} := ⟨x_{n_{j}}⟩, for j = 1, …, l, and X_{n} := ∏_{j=1}^{l} X_{n_{j}}. Note that, if l ≥ 2 then the x_{n_{j}} are squares in Q_{n}, so that X_{n} ≤ Φ(Q_{n}).

(b) When viewing the wreath product P_{n_{j}} = P_{\frac{n_{j}}{2}} ≀ C_{2} as a subgroup of 𝔖_{n_{j}} as usual, the subgroup of P_{n_{j}} corresponding to the base group is isomorphic to P_{\frac{n_{j}}{2}} × P_{\frac{n_{j}}{2}} and shall be denoted by B_{n_{j}} := P′_{n_{j}} × P′′_{n_{j}}, for j = 1, …, l. Here P′_{n_{j}} ≅ P_{\frac{n_{j}}{2}} is supposed to be acting on Ω′_{j} := \{n_{1} + ⋯ + n_{j−1} + 1, …, n_{1} + ⋯ + n_{j−1} + n_{j}/2\}, and P′′_{n_{j}} ≅ P_{\frac{n_{j}}{2}} acting on Ω′′_{j} :=

{n_{1} + ⋯ + n_{j−1} + n_{j}/2 + 1, …, n_{1} + ⋯ + n_{j}}. We also set B_{n} := ∏_{j=1}^{l} B_{n_{j}}, as well as B′_{n_{j}} := B_{n_{j}} ∩ 𝔄_{n} ≤ Q_{n_{j}}, for j = 1, …, l, and B′_{n} := B_{n} ∩ 𝔄_{n} ≤ Q_{n}. Notice that y_{n_{j}}^{2} = w_{n_{j},j}w_{n_{j−1},j} ⋯ w_{2,j}w_{n_{j},j} · w_{n_{j−1},j} ⋯ w_{2,j} ∈ B′_{n_{j}}, for j = 1, …, l.

3 The case l = 1

We investigate the case where n ≥ 8 is a 2-power.

Lemma 3.1. Let n = 2^{m}, for some m ≥ 3. Suppose that R is a maximal subgroup of Q_{n} such that E is relatively R-projective. Then, if m > 3 we have R = B_{n} ∩ Q_{n} = B′_{n}. If m = 3 then we have R ∈ {B′_{8}, (Q_{4} × Q_{4})⟨w_{8}⟩, (Q_{4} × Q_{4})⟨w_{8}^{w_{2}}⟩}.

Proof. By [13, Prop. 4.2], we know that Res_{Q_{n}}^{𝔄_{n}}(E) is indecomposable. Hence fix some vertex Q < Q_{n} of E, such that Q ≤ R. Then, by Remark 1.3, we have ⟨Φ(Q_{n}), Q_{n−4}, Q⟩ ≤ R. If n = 8 then Q_{4}Φ(Q_{8}) is a normal subgroup of Q_{8}, and Q_{8}/Q_{4}Φ(Q_{8}) is elementary abelian of order 4. Thus there are three maximal subgroups of Q_{8} containing Q_{4}Φ(Q_{8}), and these are precisely the ones listed; note that B′_{8} = (Q_{4} × Q_{4})⟨w′_{2}⟩ where w′_{2} = w_{8}^{w_{2}} · w_{8}. Hence we may now suppose that m > 3, and it suffices to prove Φ(Q_{n})Q_{n−4} = Q_{n−4}Φ(Q_{n}) = B_{n} ∩ Q_{n}. But, since m > 3, we have n − 4 = 2^{m} − 4 > 2^{m−1} + 2, and the latter assertion is immediate from Proposition 2.6 and the fact that Q_{n−4} < Q_{n}.

Proposition 3.2. Let n = 2^{m}, for some m ≥ 3. Then E has vertex Q_{n} and source Res_{Q_{n}}^{𝔄_{n}}(E).

Proof. As already mentioned, Res_{Q_{n}}^{𝔄_{n}}(E) is indecomposable, by [13, Prop. 4.2]. We follow the strategy given in Remark 1.3, and assume that R < Q_{n} is a maximal subgroup such that E is relatively R-projective.

Let first m > 3. Then, by Lemma 3.1, we have R = B_{n} ∩ Q_{n} ≥ Φ(P_{n}). In particular, ⟨x_{n}⟩ = X_{n} ≤ Φ(P_{n}) ≤ R where x_{n} is the permutation of cycle type (n/2, n/2) defined in Remark 2.7. As in the proof of [13, Prop. 4.2] we get Res_{X_{n}}^{𝔄_{n}}(E) ≅ T_{\frac{n−2}{2}} ⊕ T_{\frac{n−2}{2}} , where T_{\frac{n−2}{2}} denotes the indecomposable F X_{n}-module of dimension (n − 2)/2. Thus we conclude that Res_{R}^{𝔄_{n}}(E) = E_{1} ⊕ E_{2} where E_{1} and E_{2} are in fact uniserial with trivial heads and trivial socles. Thus, in particular, Soc(Res_{X_{n}}^{𝔄_{n}}(E)) = Res_{X_{n}}^{R}(Soc(Res_{R}^{𝔄_{n}}(E))). We set

s_{1} := ∑_{i=1}^{n/2} γ_{i} ∈ M′ and s_{2} := ∑_{i=1}^{n/4} γ_{i} + ∑_{i=1+n/2}^{3n/4} γ_{i} ∈ M′.

Then s_{1}(x_{n} + 1) = 0 and s_{2}(x_{n} + 1) = ∑_{i=1}^{n} γ_{i} ∈ M′′. Consequently, \overline{s}_{1} := s_{1} + M′′ and \overline{s}_{2} := s_{2} + M′′ are annihilated by Rad(F X_{n}), that is they are contained in Soc(Res_{X_{n}}^{𝔄_{n}}(E)). Furthermore, \overline{s}_{1} and \overline{s}_{2} are linearly independent over F. Therefore \{\overline{s}_{1}, \overline{s}_{2}\} is an F -basis for Soc(Res_{R}^{𝔄_{n}}(E)). But we also have w_{2^{m−1}} ∈ B_{n} ∩ Q_{n} = R, yielding the contradiction

\overline{s}_{2}w_{2^{m−1}} = ∑_{i=1+n/4}^{3n/4} \overline{γ}_{i} ≠ \overline{s}_{2}.

Let finally m = 3. Then R is one of the subgroups given in Lemma 3.1. To exclude the case R_{1} := B′_{8}, notice that, by Remark 2.7, we have x_{8} = y_{8}^{2} = (1, 3, 2, 4)(5, 7, 6, 8) ∈ R_{1}, and hence we may argue as in the case m > 3. As for R_{2} := (Q_{4} × Q_{4})⟨w_{8}⟩, we have w_{8}w_{4} = (1, 5, 3, 7)(2, 6, 4, 8) ∈ R_{2}, and thus letting s_{1} := γ_{1} + γ_{3} + γ_{5} + γ_{7} ∈ M′ and s_{2} := γ_{1} + γ_{3} + γ_{2} + γ_{4} ∈ M′ instead we similarly arrive at a contradiction, using s_{1}w_{4}^{w_{2}} = γ_{2} + γ_{4} + γ_{5} + γ_{7} where w_{4}^{w_{2}} = (1, 4)(2, 3) ∈ Q_{4}. For R_{3} := (Q_{4} × Q_{4})⟨w_{8}^{w_{2}}⟩ again argue similarly, using w_{8}^{w_{2}}w_{4} = (1, 6, 3, 7)(2, 5, 4, 8) ∈ R_{3}, and s_{1} := γ_{1} + γ_{3} + γ_{6} + γ_{7} ∈ M′ and s_{2} := γ_{1} + γ_{3} + γ_{2} + γ_{4} ∈ M′, where s_{1}w_{4}^{w_{2}} = γ_{2} + γ_{4} + γ_{6} + γ_{7}.

This completes the proof of Theorem 1.1 in the case where n is a 2-power.

4 The case l ≥ 2

We now investigate the case where n ≥ 4 is even, but not a 2-power. We aim to show that the vertices of E = Res_{𝔄_{n}}^{𝔖_{n}}(D) are then always the Sylow 2-subgroups of 𝔄_{n}, unless n = 6. We first need a few preparations, concerning restrictions of E to various subgroups: Recall from Remark 2.7 that Q_{n} contains the abelian subgroups Y′_{n} and X_{n}, where, since l ≥ 2, we moreover have X_{n} ≤ Φ(Q_{n}). We will investigate the restrictions of E to Y′_{n} and X_{n}, respectively, in detail. Thus we gain information on restrictions of E to Q_{n} and to maximal subgroups of Q_{n}. This leads to a further division into various subcases, which will be dealt with in the subsequent sections.

Remark 4.1. In Remark 1.2 we introduced the permutation basis {γ_{1}, …, γ_{n}} of the natural permutation F 𝔖_{n}-module M. While working with the group Y′_{n}, it will be convenient to renumber the elements of this basis as follows: for j = 1, …, l and i = n_{1} + ⋯ + n_{j−1} + 1, …, n_{1} + ⋯ + n_{j}, we set

δ_{i} := γ_{n_{1}+···+n_{j−1}+1}y_{n_{j}}^{i−n_{1}−···−n_{j−1}−1}.

In particular, for j = 1, …, l and i = n_{1} + ⋯ + n_{j−1} + 1, …, n_{1} + ⋯ + n_{j} − 1 we have δ_{i}y_{n_{j}} = δ_{i+1}, and δ_{n_{1}+···+n_{j}}y_{n_{j}} = δ_{n_{1}+···+n_{j−1}+1}. That is, {δ_{n_{1}+···+n_{j−1}+1}, …, δ_{n_{1}+···+n_{j}}} is the permutation basis of the natural permutation F Y_{n_{j}}-module, for j = 1, …, l.

For j = 1, …, l we now define

δ_{j′}^{+} := ∑_{i=0}^{n_{j}/2−1} δ_{n_{1}+···+n_{j−1}+1+2i} = ∑_{i∈Ω′_{j}} γ_{i} =: γ_{j′}^{+},

δ_{j′′}^{+} := ∑_{i=0}^{n_{j}/2−1} δ_{n_{1}+···+n_{j−1}+2+2i} = ∑_{i∈Ω′′_{j}} γ_{i} =: γ_{j′′}^{+},

and

δ_{0}^{+} := ∑_{i=0}^{n/2−1} δ_{2+2i} = ∑_{j=1}^{l} δ_{j′′}^{+} = ∑_{j=1}^{l} γ_{j′′}^{+} =: γ_{0}^{+},

as well as δ_{j}^{+} := δ_{j′}^{+} + δ_{j′′}^{+} = γ_{j′}^{+} + γ_{j′′}^{+} =: γ_{j}^{+}, and δ^{+} := ∑_{j=1}^{n} δ_{j} = ∑_{j=1}^{n} γ_{j} =: γ^{+}. The sets Ω′_{j} and Ω′′_{j} are as in Remark 2.7.

Proposition 4.2. With the above notation, for Res_{Y′_{n}}^{𝔄_{n}}(E) = Res_{Y′_{n}}^{𝔖_{n}}(M′/M′′), we have:

(i) If l > 2 then \{\overline{δ}_{j}^{+} | j ∈ \{1, …, l − 1\}\} is an F -basis of Soc(Res_{Y′_{n}}^{𝔄_{n}}(E)).

(ii) If l = 2 and n_{l} > 2 then \{\overline{δ}_{0}^{+}, \overline{δ}_{1}^{+}\} is an F -basis of Soc(Res_{Y′_{n}}^{𝔄_{n}}(E)).

(iii) If l = 2 and n_{l} = 2 then Res_{Y′_{n}}^{𝔄_{n}}(E) ≅ F Y′_{n} with socle spanned by \overline{δ}_{1}^{+}.

Proof. Let v = ∑_{i=1}^{n} c_{i}δ_{i} ∈ M, for appropriate c_{i} ∈ F, such that v̄ ∈ Soc(Res_{Y′_{n}}^{𝔖_{n}}(M/M′′)). Since the group Y′_{n} acts trivially on Soc(Res_{Y′_{n}}^{𝔖_{n}}(M/M′′)), for each j ∈ {1, …, l} there is some a_{j} ∈ F such that vy′_{n_{j}} = v + a_{j}δ^{+}.

Suppose first that l > 2 so that Ω \ (Ω_{j} ∪ Ω_{l}) ≠ ∅, for j = 1, …, l. That is, δ_{i}y′_{n_{j}} = δ_{i}, for i ∈ Ω \ (Ω_{j} ∪ Ω_{l}). In particular, for j = 1, …, l and i ∈ Ω \ (Ω_{j} ∪ Ω_{l}), we get c_{i} = c_{i} + a_{j}. Thus a_{j} = 0, for j = 1, …, l. If j ≤ l − 1 then vy′_{n_{j}} = v holds if and only if c_{n_{1}+···+n_{j−1}+1} = … = c_{n_{1}+···+n_{j}} and c_{n_{1}+···+n_{l−1}+1} = ⋯ = c_{n_{1}+···+n_{l}}. Furthermore, vy′_{n_{j}} = v for j = 1, …, l − 1 implies also vy′_{n_{l}} = v. Consequently, we deduce:

⋂_{j=1}^{l}{v ∈ M | vy′_{n_{j}} = v} = ⋂_{j=1}^{l−1}{v ∈ M | vy′_{n_{j}} = v} = ⟨δ_{1}^{+}, …, δ_{l}^{+}⟩.

Moreover, δ_{1}^{+}, …, δ_{l}^{+} are linearly independent, and thus form a basis for Soc(Res_{Y′_{n}}^{𝔖_{n}}(M)).

This in turn implies that \{\overline{δ}_{j}^{+} | j ∈ \{1, …, l − 1\}\} ⊂ M′/M′′ is an F -basis for

{v̄ ∈ M/M′′ | v ∈ ⟨δ_{1}^{+}, …, δ_{l}^{+}⟩_{F} } = {v̄ ∈ M/M′′ | v̄y′_{n_{j}} = v̄ for j = 1, …, l}
= Soc(Res_{Y′_{n}}^{𝔖_{n}}(M/M′′)).

Hence Soc(Res_{Y′_{n}}^{𝔖_{n}}(M′/M′′)) = Soc(Res_{Y′_{n}}^{𝔖_{n}}(M/M′′)) = ⟨\overline{δ}_{j}^{+} | j = 1, …, l − 1⟩_{F}, implying (i).

Next suppose that l = 2. If also j = 2 then, since Ω \ Ω_{2} = Ω_{1} ≠ ∅, we get a_{2} = 0. If j = 1 then, since Ω \ (Ω_{1} ∪ Ω_{2}) = ∅, we deduce vy′_{n_{1}} = ∑_{i=1}^{n} c_{i^{(y{′−1 \atop n_{1}})}}δ_{i} = ∑_{i=1}^{n}(c_{i} + a_{1})δ_{i}, so that

c_{i} = \begin{cases}c_{1}, & if i ∈ \{1, …, n_{1}\} is odd, \\ c_{1} + a_{1}, & if i ∈ \{1, …, n_{1}\} is even, \\ c_{n_{1}+1}, & if i ∈ \{n_{1} + 1, …, n_{1} + n_{2}\} is odd, \\ c_{n_{1}+1} + a_{1}, & if i ∈ \{n_{1} + 1, …, n_{1} + n_{2}\} is even. \end{cases}

Hence {δ_{0}^{+}, δ_{1}^{+}, δ_{2}^{+}} is an F -basis of {v ∈ M | vy′_{n_{1}} = v + a_{1}δ^{+}}. Thus the condition 0 ≠ v̄ ∈ Soc(Res_{Y′_{n}}^{𝔖_{n}}(M/M′′)) is equivalent to vy′_{n_{2}} = v where v ∈ ⟨δ_{0}^{+}, δ_{1}^{+}, δ_{2}^{+}⟩_{F}. But vy′_{n_{2}} = v holds if and only if c_{n_{1}+1} = c_{n_{1}+i}, for i = 1, …, n_{2} − 1 odd, and c_{n_{1}+2} = c_{n_{1}+i} for i = 2, …, n_{2} even. That is, {δ_{1}, …, δ_{n_{1}}, δ_{2′}^{+}, δ_{2′′}^{+}} is an F -basis of {v ∈ M | vy′_{n_{2}} = v}. Since ⟨δ_{0}^{+}, δ_{1}^{+}, δ_{2}^{+}⟩_{F} ≤ ⟨δ_{1}, …, δ_{n_{1}}, δ_{2′}^{+}, δ_{2′′}^{+}⟩_{F}, we get Soc(Res_{Y′_{n}}^{𝔖_{n}}(M/M′′)) = ⟨\overline{δ}_{0}^{+}, \overline{δ}_{1}^{+}, \overline{δ}_{2}^{+}⟩_{F}. Furthermore, \overline{δ}_{0}^{+} and \overline{δ}_{1}^{+}

are linearly independent over F, whereas \overline{δ}_{2}^{+} = \overline{δ}_{1}^{+}.

Therefore, if n_{l} > 2 then \{\overline{δ}_{0}^{+}, \overline{δ}_{1}^{+}\} ⊂ M′/M′′ is in fact a basis of Soc(Res_{Y′_{n}}^{𝔖_{n}}(M/M′′)) = Soc(Res_{Y′_{n}}^{𝔖_{n}}(M′/M′′)), proving (ii). If finally n_{l} = 2 then we have δ_{0}^{+} ∉ M′. Hence in this case

we deduce that Soc(Res_{Y′_{n}}^{𝔖_{n}}(M′/M′′)) has F -basis \{\overline{δ}_{1}^{+}\}. In particular, Res_{Y′_{n}}^{𝔄_{n}}(E) is indecomposable so that, comparing dimensions, we get Res_{Y′_{n}}^{𝔄_{n}}(E) ≅ F Y′_{n}, proving (iii).

Proposition 4.3. Let l ≥ 2 and j ∈ {1, …, l}, and let K_{j} := ker_{D}(y_{n_{j}}^{′+}) and I_{j} := im_{D}(y_{n_{j}}^{′+}).

(i) If j < l then dim(K_{j}) = dim(D) − 1, and I_{j} = ⟨\overline{δ}_{j}^{+}⟩_{F}.

(ii) If j = l and n_{l} > 2 then dim(K_{j}) = dim(D) − 2, and I_{j} = ⟨\overline{δ}_{l′}^{+}, \overline{δ}_{l′′}^{+}⟩_{F}.

(iii) If j = l and n_{l} = 2 then y_{n_{j}}^{′+} = 1 and thus K_{j} = 0.

Proof. For j = 1, …, l we define the following elements in M′:

v_{j} := δ_{n_{1}+···+n_{j}} + δ_{n_{1}+···+n_{j}+1}, if j < l,

as well as v′_{l} := δ_{n} + δ_{1} and v′′_{l} := δ_{n−1} + δ_{1}. Then v_{j}y_{n_{j}}^{′+} = δ_{j}^{+}, if j ≤ l − 1, and

v′_{l}y_{n_{l}}^{′+} = \begin{cases}v′_{l}, & if n_{l} = 2, \\ δ_{l′′}^{+}, & if n_{l} > 2, \end{cases} and v′′_{l}y_{n_{l}}^{′+} = \begin{cases}v′′_{l}, & if n_{l} = 2, \\ δ_{l′}^{+}, & if n_{l} > 2. \end{cases}

We first suppose that j < l. Then Y′_{n_{j}} acts regularly on U_{j} := ⟨δ_{n_{1}+···+n_{j−1}+1}, …, δ_{n_{1}+···+n_{j}}⟩_{F}, so that U_{j}y_{n_{j}}^{′+} ≅ Rad^{n_{j}−1}(F Y′_{n_{j}}) ≅ F and dim(ker_{U_{j}}(y_{n_{j}}^{′+})) = n_{j} − 1. Furthermore, the vector space ⟨δ_{n_{1}+···+n_{l−1}+1}, …, δ_{n_{1}+···+n_{l}}⟩_{F} is an indecomposable F Y′_{n_{j}}-module of dimension n_{l}. In particular, ⟨δ_{n_{1}+···+n_{l−1}+1}, …, δ_{n_{1}+···+n_{l}}⟩_{F} is annihilated by Rad^{n_{j}−1}(F Y′_{n_{j}}) = F ⟨y_{n_{j}}^{′+}⟩ = Soc(F Y′_{n_{j}}). Furthermore, Y′_{n_{j}} acts trivially on the subspace

⟨{δ_{1}, …, δ_{n}} \ {δ_{n_{1}+···+n_{j−1}+1}, …, δ_{n_{1}+···+n_{j}}, δ_{n_{1}+···+n_{l−1}+1}, …, δ_{n_{1}+···+n_{l}}}⟩_{F}

of M which is thus annihilated by y_{n_{j}}^{′+}. To summarize, we have shown that dim(ker_{M}(y_{n_{j}}^{′+})) = n − 1 = dim(M′). Hence dim(ker_{M′}(y_{n_{j}}^{′+})) ∈ {dim(M′), dim(M′) − 1} = {n − 1, n − 2} and dim(ker_{D}(y_{n_{j}}^{′+})) = dim(K_{j}) ∈ {dim(D), dim(D) − 1} = {n − 2, n − 3}. But we already know

that 0 ≠ \overline{δ}_{j}^{+} ∈ im_{D}(y_{n_{j}}^{′+}) = I_{j} so that I_{j} = ⟨\overline{δ}_{j}^{+}⟩_{F} and dim(K_{j}) = dim(D) − 1, proving (i).

Now let j = l and n_{l} > 2. Then Y′_{n_{l}} acts trivially on ⟨δ_{1}, …, δ_{n_{1}+...+n_{l−1}}⟩_{F}. Furthermore, ⟨δ_{n_{1}+···+n_{l−1}+1}, …, δ_{n_{1}+...+n_{l}}⟩_{F} is an F Y′_{n_{l}}-module isomorphic to F Y′_{n_{l}} ⊕ F Y′_{n_{l}}. Consequently, ⟨δ_{1}, …, δ_{n_{1}+...+n_{l−1}}⟩_{F} is annihilated by y_{n_{l}}^{′+}, and

⟨δ_{n_{1}+···+n_{l−1}+1}, …, δ_{n_{1}+...+n_{l}}⟩_{F} · y_{n_{l}}^{′+} ≅ Rad^{n_{l}−1}(F Y′_{n_{l}}) ⊕ Rad^{n_{l}−1}(F Y′_{n_{l}}) ≅ F ⊕ F.

Thus dim(ker_{M}(y_{n_{l}}^{′+})) = n − 2 = dim(M′) − 1 so that dim(ker_{M′}(y_{n_{l}}^{′+})) ≥ dim(M′) − 2 = n − 3 and dim(K_{l}) = dim(ker_{D}(y_{n_{l}}^{′+})) ≥ dim(D) − 2 = n − 4. As we have already shown above, ⟨\overline{δ}_{l′}^{+}, \overline{δ}_{l′′}^{+}⟩_{F} ≤ im_{D}(y_{n_{l}}^{′+}) = I_{l}. Since l ≥ 2, we conclude that \overline{δ}_{l′}^{+} and \overline{δ}_{l′′}^{+} are linearly independent over F, and hence I_{l} = ⟨\overline{δ}_{l′}^{+}, \overline{δ}_{l′′}^{+}⟩_{F} and dim(K_{l}) = n − 4 = dim(D) − 2. This proves (ii), and assertion (iii) is obviously true, since y′_{n_{l}} = 1 for n_{l} = 2.

Remark 4.4. Let l ≥ 2. We consider the subgroup X_{n} = ∏_{j=1}^{l} X_{n_{j}} ≤ Φ(Q_{n}) ≤ Q_{n} ≤ P_{n}. Letting Q ≤ P_{n} be any subgroup such that X_{n} ≤ Q, we show when X_{n} can be used to prove the indecomposability of Res_{Q}^{𝔖_{n}}(D). Note that this is slightly more general than needed to prove Theorem 1.1, but will be useful in the proof of Theorem 9.1. Therefore, we fix an indecomposable direct sum decomposition of Res_{Q}^{𝔖_{n}}(D).

(a) Suppose that j ∈ {1, …, l} such that n_{j} > 2, that is, we exclude just the case j = l and n_{l} = 2. Let u := γ_{n_{1}+···+n_{j}} +γ_{n_{1}+···+n_{j}+1} ∈ M′ and v := γ_{n_{1}+···+n_{j−1}+1} +γ_{n_{1}+···+n_{j−1}+n_{j}/2+1} ∈ M′ where the indices should be read modulo n. Then

ux_{n_{j}}^{+} = γ_{j′′}^{+} = δ_{j′′}^{+} ∉ M′′ and vx_{n_{j}}^{+} = γ_{j}^{+} = δ_{j}^{+} ∉ M′′.  (4)

Furthermore, ⟨γ_{n_{1}+···+n_{j−1}+1}, …, γ_{n_{1}+···+n_{j}}⟩_{F} is an F X_{n_{j}}-module isomorphic to F X_{n_{j}}⊕F X_{n_{j}}, and X_{n_{j}} acts trivially on ⟨γ_{1}, …, γ_{n_{1}+···+n_{j−1}}, γ_{n_{1}+···+n_{j}+1}, …, γ_{n}⟩_{F}. Hence

im_{M}(x_{n_{j}}^{+}) = Soc(⟨γ_{n_{1}+···+n_{j−1}+1}, …, γ_{n_{1}+···+n_{j}}⟩_{F}) ≅ F ⊕ F,

and dim(ker_{M}(x_{n_{j}}^{+})) = n − 2. This implies dim(ker_{M′}(x_{n_{j}}^{+})) ≥ dim(M′) − 2 and thus dim(ker_{D}(x_{n_{j}}^{+})) ≥ dim(D) − 2. Since, by (4), \overline{γ}_{j′}^{+}, \overline{γ}_{j′′}^{+} ∈ im_{D}(x_{n_{j}}^{+}), we get dim(ker_{D}(x_{n_{j}}^{+})) = dim(D) − 2 = n − 4 and im_{D}(x_{n_{j}}^{+}) = ⟨\overline{γ}_{j′}^{+}, \overline{γ}_{j′′}^{+}⟩_{F}. Hence we have (F X_{n_{j}} ⊕ F X_{n_{j}})| Res_{X_{n_{j}}}^{𝔖_{n}} (D), and either precisely one or precisely two indecomposable summands in the fixed direct sum decomposition of Res_{Q}^{𝔖_{n}}(D) are not annihilated by x_{n_{j}}^{+}.

(b) Suppose that there are two of these, U′ and U′′, say. Recall further that we are still assuming n_{j} > 2. Then there are some a′, a′′, b′, b′′ ∈ F such that 0 ≠ u′ := a′\overline{γ}_{j′}^{+} + a′′\overline{γ}_{j′′}^{+} ∈ U′ and 0 ≠ u′′ := b′\overline{γ}_{j′}^{+} + b′′\overline{γ}_{j′′}^{+} ∈ U′′. In particular, u′ and u′′ are linearly independent. Next consider Q^{p_{j}q_{j}}, where

p_{j} : P_{n} ⟶ P_{n_{j}} and q_{j} : P_{n_{j}} ⟶ P_{n_{j}}/B_{n_{j}} ≅ ⟨w_{n_{j},j}⟩

are the natural epimorphisms. We have either Q^{p_{j}q_{j}} = ⟨w_{n_{j},j}⟩ or Q^{p_{j}q_{j}} = {1}. Assume that Q^{p_{j}q_{j}} = ⟨w_{n_{j},j}⟩. Then there exists some g ∈ Q such that γ_{j′}^{+}g = γ_{j′′}^{+} and γ_{j′′}^{+}g = γ_{j′}^{+}. If a′ ≠ a′′ then u′ + u′g = (a′ + a′′)\overline{γ}_{j}^{+} ∈ U′ and thus also \overline{γ}_{j}^{+} ∈ U′. If a′ = a′′ then 0 ≠ a′\overline{γ}_{j}^{+} ∈ U′, and again \overline{γ}_{j}^{+} ∈ U′. But, analogously, we deduce \overline{γ}_{j}^{+} ∈ U′′, yielding the contradiction 0 ≠ \overline{γ}_{j}^{+} ∈ U′ ∩ U′′. Consequently, this forces Q^{p_{j}q_{j}} = \{1\}, that is Q ≤ ker(p_{j}q_{j}).

(c) Now suppose that n_{l} > 2 and that, for each j = 1, …, l, there is exactly one indecomposable direct summand U_{j} in the fixed direct sum decomposition of Res_{Q}^{𝔖_{n}}(D) which is not annihilated by x_{n_{j}}^{+}. We thus deduce \overline{γ}_{j}^{+} ∈ ⟨\overline{γ}_{j′}^{+}, \overline{γ}_{j′′}^{+}⟩_{F} = im_{D}(x_{n_{j}}^{+}) ≤ U_{j}, for j = 1, …, l. Since ∑_{j=1}^{l} \overline{γ}_{j}^{+} = \overline{γ}^{+} = 0, this implies U_{1} = … = U_{l}.

Next we show that Soc(Res_{X_{n}}^{𝔖_{n}}(D)) ≤ Res_{X_{n}}^{Q}(U_{1}) + ⋯ + Res_{X_{n}}^{Q}(U_{l}) = Res_{X_{n}}^{Q}(U_{1}), which then implies that Res_{Q}^{𝔖_{n}}(D) = U_{1} is indecomposable. For this, let v ∈ M′. Then \overline{v} ∈ Soc(Res_{X_{n}}^{𝔖_{n}}(D)) if and only if \overline{v}x_{n_{j}} = \overline{v}, for all j = 1, …, l, that is, if and only if for each j ∈ {1, …, l} there is some a_{j} ∈ F such that vx_{n_{j}} = v + a_{j}γ^{+}. Let j ∈ {1, …, l}. Since X_{n_{j}} acts trivially on Ω \ Ω_{j} ≠ ∅, this condition is actually equivalent to vx_{n_{j}} = v. That is, \overline{v} ∈ D is fixed by X_{n_{j}} if and only v ∈ M′ is. Since ⟨γ_{n_{1}+···+n_{j−1}+1}, …, γ_{n_{1}+···+n_{j}}⟩_{F} is isomorphic to F X_{n_{j}} ⊕ F X_{n_{j}} as an F X_{n_{j}}-module, we get

Soc(Res_{X_{n}}^{𝔖_{n}}(M)) = {v ∈ M | vx_{n_{j}} = v for all j = 1, …, l} = ⨁_{j=1}^{l}⟨γ_{j′}^{+}, γ_{j′′}^{+}⟩_{F} ≤ Res_{X_{n}}^{𝔖_{n}}(M′).

Consequently, Soc(Res_{X_{n}}^{𝔖_{n}}(M)) = Soc(Res_{X_{n}}^{𝔖_{n}}(M′)) = ⨁_{j=1}^{l}⟨γ_{j′}^{+}, γ_{j′′}^{+}⟩_{F}, and thus indeed

Soc(Res_{X_{n}}^{𝔖_{n}}(D)) = ⟨\overline{γ}_{1′}^{+}, \overline{γ}_{1′′}^{+}, …, \overline{γ}_{l′}^{+}, \overline{γ}_{l′′}^{+}⟩_{F} ≤ Res_{X_{n}}^{Q}(U_{1}) + ⋯ + Res_{X_{n}}^{Q}(U_{l}).

5 The case l ≥ 2 and n_{l} > 2

The behaviour of the natural F 𝔄_{n}-module E upon restriction to the abelian subgroup Y′_{n} of Q_{n} depends on whether n_{l} > 2 or n_{l} = 2. Therefore, we will now distinguish between these two cases, and start off with the case n_{l} > 2.

Proposition 5.1. Let l ≥ 2 and n_{l} > 2. Then E has vertex Q_{n} and source Res_{Q_{n}}^{𝔄_{n}}(E).

Proof. We follow the strategy given in Remark 1.3, and assume that R < Q_{n} is a maximal subgroup such that E is relatively R-projective, hence X_{n} = ∏_{j=1}^{l} X_{n_{j}} ≤ Φ(Q_{n}) ≤ R and Q_{n−4} ≤ R.

First of all, we fix an indecomposable direct sum decomposition of Res_{Q_{n}}^{𝔄_{n}}(E). Then, for each j = 1, …, l, we have Q_{n}^{p_{j}q_{j}} = ⟨w_{n_{j},j}⟩ ≠ 1; note that here we need n_{l} ≥ 4. Therefore, in consequence of Remark 4.4, part (b), we obtain that, for each j = 1, …, l, there is precisely one indecomposable direct summand U_{j} in the decomposition which is not annihilated by x_{n_{j}}^{+}. Hence, Remark 4.4, part (c), shows that Res_{Q_{n}}^{𝔄_{n}}(E) is, in fact, indecomposable.

Now we fix an indecomposable direct sum decomposition of Res_{R}^{𝔄_{n}}(E). Hence, by Remark 4.4, part (a), there is either one indecomposable direct summand U or there are exactly two indecomposable direct summands U′ and U′′ in the decomposition which are not annihilated by x_{n_{1}}^{+}. In the first case we get (F X_{n_{1}} ⊕ F X_{n_{1}})| Res_{X_{n_{1}}}^{R} (U), hence dim(U) ≥ 2 dim(F X_{n_{1}}) = n_{1} ≥ (n+2)/2 > (n−2)/2 = dim(E)/2, a contradiction. In the second case Remark 4.4, part (b), implies that R ≤ ker(p_{1}q_{1}). But, since l ≥ 2 and n_{l} ≥ 4, we have w_{n_{1},1} ∈ Q_{n_{1}} ≤ Q_{n−4} ≤ R and w_{n_{1},1} ∉ ker(p_{1}q_{1}), a contradiction.

It remains to deal with the case n_{l} = 2. It turns out that E behaves differently upon restriction to Y′_{n}, depending on the value of l ≥ 2. In the following section we begin with the generic case l ≥ 4, while the exceptional cases l ≤ 3 will be settled in subsequent sections.

6 The case n_{l} = 2 and l ≥ 4

Lemma 6.1. Let n_{l} = 2 and l ≥ 4. Then Res_{Y′_{n}}^{𝔄_{n}}(E) is indecomposable.

Proof. We fix an indecomposable direct sum decomposition of Res_{Y′_{n}}^{𝔄_{n}}(E). For j ∈ {1, …, l − 1}, by Proposition 4.3, there is precisely one indecomposable direct summand V_{j} in this decomposition which is not annihilated by y_{n_{j}}^{′+}. We denote the preimage of V_{j} under the natural epimorphism M′ ⟶ M′/M′′ by V̂_{j}. From \overline{γ}_{j}^{+} ∈ V_{j}, for j = 1, …, l − 1, we by Proposition 4.2 conclude that Soc(Res_{Y′_{n}}^{𝔄_{n}}(E)) ≤ V_{1} + ⋯ + V_{l−1}, so that Res_{Y′_{n}}^{𝔄_{n}}(E) = V_{1} + ⋯ + V_{l−1}. Hence it suffices to show that V_{1} = … = V_{l−1}. Suppose, for a contradiction, that this is not the case. After appropriate re-numbering, we may suppose that V_{1} = … = V_{k} and V_{k+1} ≠ V_{1}, …, V_{l−1} ≠ V_{1}. Furthermore, since l ≥ 4, we may also suppose that k < l − 2.

For i = 1, …, l, we have the F P_{n}-epimorphism

π_{i} : M ⟶ M_{n_{i}} := ⟨γ_{n_{1}+···+n_{i−1}+1}, …, γ_{n_{1}+···+n_{i}}⟩_{F} ,

and we consider π := ⨁_{j=1}^{k} π_{j} ⊕ π_{l} : M ⟶ ⨁_{j=1}^{k} M_{n_{j}} ⊕ M_{n_{l}}. For j = 1, …, l and v ∈ M_{n_{j}}, set h_{j}(v) := min{e ∈ ℕ_{0} | v ∈ ker_{M_{n_{j}}} ((y_{n_{j}} + 1)^{e})}; note that h_{j}(v) = min{e ∈ ℕ_{0} | v ∈ ker_{M_{n_{j}}} ((y′_{n_{j}} + 1)^{e})} whenever j ≤ l − 1.

Now assume that there is some 0 ≠ ṽ ∈ V̂_{1} ∩ker(π). If h_{j}(ṽ^{π_{j}}) ≤ 1 for all j ∈ {k+1, …, l−1}, then ṽ^{π_{j}} = a_{j}γ_{j}^{+}, for some a_{j} ∈ F, and hence 0 ≠ ṽ ∈ V̂_{1} ∩ (V̂_{k+1} + ⋯ + V̂_{l−1}) = ⟨γ^{+}⟩_{F}. If there is some j ∈ {k + 1, …, l − 1} such that h_{j}(ṽ^{π_{j}}) > 1, then let v := ṽ(y′_{n_{j}} + 1)^{h_{j}(ṽ^{π_{j}})−1} ∈ (V̂_{1} ∩ ker(π)) \ {0}, hence v^{π_{r}} = 0 for r ∈ {k + 1, …, l − 1} \ {j}, and v^{π_{j}} = aγ_{j}^{+}, for some 0 ≠ a ∈ F, thus 0 ≠ v ∈ V̂_{1} ∩ V̂_{j} = ⟨γ^{+}⟩_{F}. In both cases, γ^{+} ∉ ker(π) yields a contradiction. Consequently, π_{|V̂_{1}} is injective.

Next we consider the F P_{n}-epimorphism π′ := ⨁_{j=1}^{k} π_{j} : M ⟶ ⨁_{j=1}^{k} M_{n_{j}}. Assume that there is some 0 ≠ ṽ ∈ V̂_{1} ∩ ker(π′). We write ṽ = ∑_{i=1}^{n} a_{i}δ_{i}, for appropriate a_{1}, …, a_{n} ∈ F. By the injectivity of π_{|V̂_{1}} we have ṽ ∉ ker(π), that is, a_{n−1} ≠ 0 or a_{n} ≠ 0. We set v_{0} := ∑_{j=k+1}^{l} γ_{j}^{+} = ∑_{j=k+1}^{l} δ_{j}^{+}, then v_{0} ∈ ker(π′). Moreover, v_{0} = γ^{+} + ∑_{j=1}^{k} γ_{j}^{+}, and thus also v_{0} ∈ V̂_{1}. If a_{n−1} = a_{n} ≠ 0, then we may suppose that a_{n−1} = 1 = a_{n}, and thus

ṽ + v_{0} = ∑_{j=n_{1}+···+n_{k}+1}^{n_{1}+···+n_{l−1}}(a_{j} + 1)δ_{j} ∈ V̂_{1} ∩ ker(π) = {0},

hence ṽ = ∑_{j=k+1}^{l} γ_{j}^{+} = v_{0}. Now assume that a_{n−1} ≠ a_{n}. Let j ∈ {k + 1, …, l − 1}. Then

0 ≠ v := ṽ(y′_{n_{j}} + 1) = (  ∑_{i=n_{1}+···+n_{j−1}+2}^{n_{1}+···+n_{j}}(a_{i} + a_{i−1})δ_{i})

+ (a_{n_{1}+···+n_{j−1}+1} + a_{n_{1}+···+n_{j}})δ_{n_{1}+···+n_{j−1}+1} + (a_{n−1} + a_{n})(δ_{n−1} + δ_{n}) ∈ V̂_{1} ∩ ker(π′).

Hence we have v = bv_{0}, for some 0 ≠ b ∈ F, which in turn means that j = k + 1 = l − 1. But, since k < l − 2, this is a contradiction. Hence we actually have a_{n−1} = a_{n} ≠ 0, and thus V̂_{1} ∩ ker(π′) = ⟨v_{0}⟩_{F}.

We now show that π′_{|V̂_{1}} is surjective: For j ∈ {k+1, …, l−1} and i ∈ {1, …, k} we have V̂_{j}^{π_{i}} ≤

Rad(Res_{Y′_{n_{i}}}^{P_{n}} (M_{n_{i}})), since otherwise taking w ∈ V̂_{j} such that w^{π_{i}} ∈ M_{n_{i}} \ Rad(Res_{Y′_{n_{i}}}^{P_{n}} (M_{n_{i}}))

yields w(y′_{n_{i}} + 1)^{n_{i}−1} = aγ_{i}^{+} ∈ V̂_{j}, for some 0 ≠ a ∈ F, a contradiction. Hence we have

(V̂_{k+1} + ⋯ + V̂_{l−1})^{π′} ≤ ⨁_{i=1}^{k} Rad(Res_{Y′_{n_{i}}}^{P_{n}} (M_{n_{i}})) = Rad(Res_{Y′_{n}}^{P_{n}}(⨁_{i=1}^{k} M_{n_{i}})).

From V̂_{1} + ⋯ + V̂_{l−1} = Res_{Y′_{n}}^{𝔖_{n}}(M′) we get (V̂_{1} + V̂_{k+1} + ⋯ + V̂_{l−1})^{π′} = Res_{Y′_{n}}^{P_{n}}(⨁_{i=1}^{k} M_{n_{i}}), hence V̂_{1}^{π′} = Res_{Y′_{n}}^{P_{n}}(⨁_{i=1}^{k} M_{n_{i}}).

Thus there is some w ∈ V̂_{1} such that w^{π′} = γ_{1} = δ_{1}. We write w = δ_{1} + ∑_{i=n_{1}+···+n_{k}+1}^{n} a_{i}δ_{i}, for some a_{n_{1}+···+n_{k}+1}, …, a_{n} ∈ F. Furthermore, ∑_{i=n_{1}+···n_{k}+1}^{n} a_{i} = 1, since w ∈ V̂_{1} ≤ M′. Assume there is j ∈ {k + 1, …, l − 1} such that h_{j}(w^{π_{j}}) > 1. Then let

w̃ := w(y′_{n_{j}} + 1) ∈ (V̂_{1} ∩ ker(π′)) \ {0}.

Hence we have w̃^{π_{j}} ≠ 0, and w̃^{π_{r}} = 0 for j ≠ r ∈ {k + 1, …, l − 1}, while w̃^{π_{l}} = (a_{n−1} + a_{n})(δ_{n−1} + δ_{n}). But V̂_{1} ∩ ker(π′) = ⟨v_{0}⟩_{F} and our hypothesis k < l − 2 yield a contradiction, thus we have h_{j}(w^{π_{j}}) ≤ 1 for all j ∈ {k + 1, …, l − 1}. In particular, ∑_{i=n_{1}+···+n_{j−1}+1}^{n_{1}+···+n_{j}} a_{i} = 0, for j ∈ {k + 1, …, l − 1}. This forces a_{n−1} + a_{n} = 1, that is h_{l}(w^{π_{l}}) = 2. Thus, for any j ∈ {k + 1, …, l − 1}, we get w(y′_{n_{j}} + 1) = (a_{n−1} + a_{n})(δ_{n−1} + δ_{n}) ∈ (V̂_{1} ∩ ker(π′)) \ {0}, contradicting V̂_{1} ∩ ker(π′) = ⟨v_{0}⟩_{F}.

Proposition 6.2. Let n_{l} = 2 and l ≥ 4. Then E has vertex Q_{n} and source Res_{Q_{n}}^{𝔄_{n}}(E).

Proof. We follow the strategy given in Remark 1.3. By Lemma 6.1, we already know that Res_{Q_{n}}^{𝔄_{n}}(E) is indecomposable. Assume that R < Q_{n} is a maximal subgroup such that E is relatively R-projective, hence X_{n} = ∏_{j=1}^{l} X_{n_{j}} ≤ Φ(Q_{n}) ≤ R and Q_{n−4} ≤ R. We fix an indecomposable direct sum decomposition of Res_{R}^{𝔄_{n}}(E). By Remark 4.4, part (a), there is either exactly one summand U or there are exactly two summands U′ and U′′ in this decomposition which are not annihilated by x_{n_{1}}^{+}. In the first case (F X_{n_{1}} ⊕ F X_{n_{1}})| Res_{X_{n_{1}}}^{R} (U) implies dim(U) ≥ n_{1} > dim(E)/2, a contradiction. In the second case, Remark 4.4, part (b), implies that R ≤ ker(p_{1}q_{1}). But since l ≥ 3 and n_{l} = 2 we have Q_{n−4} = (P_{n_{1}} × ⋯ × P_{n_{l−2}} × P_{2^{i_{l−1}−1}} × P_{2^{i_{l−1}−2}} × ⋯ × P_{4} × P_{2}) ∩ Q_{n}, thus w_{n_{1},1} ∈ Q_{n_{1}} ≤ Q_{n−4} ≤ R, a contradiction.

7 The case n_{l} = 2 and l = 3

Lemma 7.1. Let n_{l} = 2 and l = 3. Then Res_{Y′_{n}}^{𝔄_{n}}(E) = U_{1} ⊕ U_{2} where U_{1} and U_{2} are indecomposable of dimension n_{1} and n_{2}, respectively, and both have vertex Y′_{n}.

Proof. We construct the following subspaces M_{1} and M_{2} of M′: M_{1} has F -basis 𝔅_{1} := {b_{1}, …, b_{n_{1}}} with

b_{1} := δ_{1} + δ_{n−1} + ∑_{i=0}^{n_{2}/2−1} δ_{n_{1}+1+2i} and b_{j} := δ_{j} + δ_{j−1} + δ_{n−1} + δ_{n},

for j = 2, …, n_{1}. Obviously, 𝔅_{1} is indeed linearly independent. Moreover, M_{1} + ⟨δ^{+}⟩_{F} is an F Y′_{n}-module, since

b_{j}y′_{n_{1}} = \begin{cases}b_{1} + b_{2}, & for j = 1, \\ b_{j+1}, & for j = 2, …, n_{1} − 1, \\ ∑_{i=2}^{n_{1}} b_{i}, & for j = n_{1}, \end{cases} b_{j}y′_{n_{2}} = \begin{cases}b_{1} + ∑_{i=1}^{n_{1}/2} b_{2i} + δ^{+}, & for j = 1, \\ b_{j}, & for j = 2, …, n_{1}. \end{cases}

Similarly, we define the subspace M_{2} of M′ with basis 𝔅_{2} := {b̃_{1}, …, b̃_{n_{2}}} where

b̃_{1} := δ_{n_{1}+1} + δ_{n−1} + ∑_{i=0}^{n_{1}/2−1} δ_{1+2i} and b̃_{j} := δ_{n_{1}+j} + δ_{n_{1}+j−1} + δ_{n−1} + δ_{n},

for j = 2, …, n_{2}. Also M_{2} + ⟨δ^{+}⟩_{F} is an F Y′_{n}-module and, by construction, M_{1} + M_{2} + ⟨δ^{+}⟩_{F} is an F Y′_{n}-submodule of Res_{Y′_{n}}^{𝔖_{n}}(M′). We show that M_{1} + M_{2} + ⟨δ^{+}⟩_{F} contains an F -basis of M′, implying that we actually have M_{1} +M_{2} +⟨δ^{+}⟩_{F} = Res_{Y′_{n}}^{𝔖_{n}}(M′): First of all, we have δ_{1}^{+} =

∑_{i=1}^{n_{1}/2} b_{2i} ∈ M_{1} and δ_{2}^{+} = ∑_{i=1}^{n_{2}/2} b̃_{2i} ∈ M_{2}. Thus δ_{n−1} +δ_{n} = δ_{1}^{+} +δ_{2}^{+} +δ^{+} ∈ M_{1} +M_{2} +⟨δ^{+}⟩_{F}, and

δ_{i} + δ_{i+1} = \begin{cases}b_{i+1} + δ_{n−1} + δ_{n} ∈ M_{1} + M_{2} + ⟨δ^{+}⟩_{F} , & for i = 1, …, n_{1} − 1, \\ b̃_{i+1} + δ_{n−1} + δ_{n} ∈ M_{1} + M_{2} + ⟨δ^{+}⟩_{F} , & for i = n_{1} + 1, …, n_{1} + n_{2} − 1. \end{cases}

Hence we have s_{1} := ∑_{i=0}^{n_{2}/2−1} δ_{n_{1}+1+2i} ∈ M_{1} + M_{2} + ⟨δ^{+}⟩_{F} and s_{2} := ∑_{i=0}^{n_{1}/2−1} δ_{1+2i} ∈ M_{1} + M_{2} + ⟨δ^{+}⟩_{F}, thus we get δ_{1} + δ_{n−1} = b_{1} + s_{1} ∈ M_{1} + M_{2} + ⟨δ^{+}⟩_{F} and δ_{n_{1}+1} + δ_{n−1} = b̃_{1} + s_{2} ∈ M_{1} + M_{2} + ⟨δ^{+}⟩_{F}, therefore,

Res_{Y′_{n}}^{𝔄_{n}}(E) = Res_{Y′_{n}}^{𝔖_{n}}(M′/M′′) = (M_{1} + M′′/M′′) + (M_{2} + M′′/M′′).

Letting U_{i} := M_{i} + M′′/M′′, we from dim(E) = n − 2 = n_{1} + n_{2} = dim(M_{1}) + dim(M_{2}) get U_{i} ≅ M_{i}, for i ∈ {1, 2}, and hence Res_{Y′_{n}}^{𝔄_{n}}(E) = U_{1} ⊕ U_{2}. For j = 1, …, n_{1} we set

\overline{b}_{j} := b_{j} + M′′, and for j = 1, …, n_{2} we set \overline{b̃}_{j} := b̃_{j} + M′′. Then \overline{𝔅}_{1} := \{\overline{b}_{1}, …, \overline{b}_{n_{1}}\} and \overline{𝔅}_{2} := \{\overline{b̃}_{1}, …, \overline{b̃}_{n_{2}}\} are bases for U_{1} and U_{2}, respectively. Furthermore, both U_{1} and U_{2} are indecomposable. Namely, b_{1}y_{n_{1}}^{′+} = δ_{1}^{+} = ∑_{i=1}^{n_{1}/2} b_{2i} ≠ 0 so that M_{1} is not annihilated by y_{n_{1}}^{′+}. In particular, F Y′_{n_{1}} | Res_{Y′_{n_{1}}}^{Y′_{n}} (M_{1}). Comparing dimensions, we deduce F Y′_{n_{1}} ≅ Res_{Y′_{n_{1}}}^{Y′_{n}} (M_{1}).

In particular, Res_{Y′_{n_{1}}}^{Y′_{n}} (M_{1}) and thus also M_{1} ≅ U_{1} is uniserial, hence indecomposable. The indecomposability of U_{2} is proved analogously.

Next we show that Y′_{n} is a vertex of both U_{1} and U_{2}. For this, notice that Y′_{n} possesses precisely three maximal subgroups, these are Z_{1} := ⟨y′_{n_{1}}, (y′_{n_{2}})^{2}⟩, Z_{2} := ⟨y′_{n_{1}}y′_{n_{2}}, (y′_{n_{2}})^{2}⟩ and Z_{3} := ⟨(y′_{n_{1}})^{2}, y′_{n_{2}}⟩. We show that neither U_{1} nor U_{2} can be relatively Z_{i}-projective, for i = 1, 2, 3. By Green’s Indecomposability Theorem, it suffices to verify that U_{1} and U_{2} restrict indecomposably to each of these groups. We investigate U_{1} first. By definition, (y′_{n_{2}})^{2} acts trivially on U_{1}. That is, we may view U_{1} as a module for the factor group \overline{Y} ′_{n} := Y′_{n}/⟨(y′_{n_{2}})^{2}⟩ and show that it restricts indecomposably to each of the maximal subgroups of \overline{Y} ′_{n}. The latter are in natural bijection with the maximal subgroups of Y′_{n}, thus are \overline{Z}_{1} = ⟨\overline{y}′_{n_{1}}⟩, \overline{Z}_{2} = ⟨\overline{y}′_{n_{1}} \overline{y}′_{n_{2}}⟩ and \overline{Z}_{3} = ⟨(\overline{y}′_{n_{1}})^{2}, \overline{y}′_{n_{2}}⟩, where ^{−} : Y′_{n} ⟶ \overline{Y} ′_{n} denotes the natural epimorphism.

As we have already mentioned, Res_{Y′_{n_{1}}}^{Y′_{n}} (U_{1}) ≅ F Y′_{n_{1}}. Hence also Res\begin{matrix}\overline{Y} & {′ \atop n} \\ \overline{Z} & 1 \end{matrix}(U_{1}) ≅ F \overline{Z}_{1} which is indecomposable.

We consider Res\begin{matrix}\overline{Y} & {′ \atop n} \\ \overline{Z} & 2 \end{matrix}(U_{1}) and Res\begin{matrix}\overline{Y} & {′ \atop n} \\ \overline{Z} & 3 \end{matrix}(U_{1}). For convenience, we replace the F -basis \overline{𝔅}_{1} =

\{\overline{b}_{1}, …, \overline{b}_{n_{1}}\} of U_{1} by \overline{𝔅}′_{1} := \{\overline{b}′_{1}, …, \overline{b}′_{n_{1}}\} where \overline{b}′_{1} := \overline{b}_{1}, and \overline{b}′_{j} := \overline{b}′_{j−1}+\overline{b}_{j}, for j = 2, …, n_{1}. In other words,

\overline{b}′_{j} = \overline{δ}_{j} + ∑_{i=0}^{n_{2}/2−1} \overline{δ}_{n_{1}+1+2i} + \begin{cases}\overline{δ}_{n−1}, & if j is odd, \\ \overline{δ}_{n}, & if j is even. \end{cases}

With this notation, we get

\overline{b}′_{j}\overline{y}′_{n_{1}} = \overline{b}′_{j}y′_{n_{1}} = \overline{b}′_{j+1}, for j = 1, …, n_{1} − 1,
\overline{b}′_{n_{1}} \overline{y}′_{n_{1}} = \overline{b}′_{n_{1}}y′_{n_{1}} = \overline{b}′_{1},
\overline{b}′_{j}\overline{y}′_{n_{2}} = \overline{b}′_{j}y′_{n_{2}} = ∑_{i≠j} \overline{b}′_{i}, for j = 1, …, n_{1}.

We set \overline{Y} ′′_{n_{1}} := ⟨\overline{y}_{n_{1}}^{′2}⟩, and notice that \overline{Y} ′′_{n_{1}} ≤ \overline{Z}_{2} and \overline{Y} ′′_{n_{1}} ≤ \overline{Z}_{3}. Since Res\begin{matrix}\overline{Y} & {′ \atop n} \\ \overline{Y} & {′ \atop n_{1}} \end{matrix} (U_{1}) ≅ F \overline{Y} ′_{n_{1}},

we also have Res\begin{matrix}\overline{Y} & {′ \atop n} \\ \overline{Y} & {′′ \atop n_{1}} \end{matrix} (U_{1}) ≅ F \overline{Y} ′′_{n_{1}} ⊕ F \overline{Y} ′′_{n_{1}}. More precisely, Res\begin{matrix}\overline{Y} & {′ \atop n} \\ \overline{Y} & {′′ \atop n_{1}} \end{matrix} (U_{1}) = V_{1} ⊕ V_{2} where

V_{1} := ⟨\overline{b}′_{i} | i odd ⟩_{F} and V_{2} := ⟨\overline{b}′_{i} | i even ⟩_{F}, and the maps V_{1} ⟶ F \overline{Y} ′′_{n_{1}}, \overline{b}′_{2i−1} ⟼ \overline{y}_{n_{1}}^{′i} and V_{2} ⟶ F \overline{Y} ′′_{n_{1}}, \overline{b}′_{2i} ⟼ \overline{y}_{n_{1}}^{′i}, for i = 1, …, n_{1}/2, are isomorphisms of F \overline{Y} ′′_{n_{1}}-modules. Consequently,

E := End_{F \overline{Y} ′′_{n_{1}}} (Res\begin{matrix}\overline{Y} & {′ \atop n} \\ \overline{Y} & {′′ \atop n_{1}} \end{matrix} (U_{1})) ≅ Mat(2, F \overline{Y} ′′_{n_{1}}),

and we will from now on simply identify E and Mat(2, F \overline{Y} ′′_{n_{1}}). Moreover, E is a \overline{Y} ′_{n}-algebra with respect to the conjugation action induced by the natural embedding F \overline{Y} ′_{n} ⟶ E. Via this embedding, the elements \overline{y}_{n_{1}}^{′2} and \overline{y}′_{n_{1}} correspond to the endomorphisms

(\begin{matrix}\overline{y}_{n_{1}}^{′2} & 0 \\ 0 & \overline{y}_{n_{1}}^{′2} \end{matrix}) and (\begin{matrix}0 & 1 \\ \overline{y}_{n_{1}}^{′2} & 0 \end{matrix}) ,

respectively. Now \overline{y}′_{n_{2}} acts on U_{1} as 1 + (\overline{y}′_{n_{1}})^{+} = 1 + (\overline{y}_{n_{1}}^{′2})^{+} + \overline{y}′_{n_{1}} · (\overline{y}_{n_{1}}^{′2})^{+}. Letting s := (\overline{y}_{n_{1}}^{′2})^{+} ∈ Soc(F \overline{Y} ′′_{n_{1}}), the elements \overline{y}′_{n_{2}} and \overline{y}′_{n_{1}} \overline{y}′_{n_{2}} correspond to the endomorphisms

ψ := (\begin{matrix}1 + s & s \\ s & 1 + s \end{matrix}) and ψ′ := (\begin{matrix}s & 1 + s \\ \overline{y}_{n_{1}}^{′2} + s & s \end{matrix}) ,

respectively.

We now determine the elements in E which are fixed by \overline{y}′_{n_{2}}. These are precisely the elements

in End_{F \overline{Z}_{3}}(U_{1}). Let φ ∈ E such that φ = (\begin{matrix}a & b \\ c & d \end{matrix}), for some a, b, c, d ∈ F \overline{Y} ′′_{n_{1}}. Then φ ∈

End_{F \overline{Z}_{3}}(U_{1}) if and only if φψ = ψφ, or equivalently, if (a + d)s = 0 = (b + c)s. That is,

F := End_{F \overline{Z}_{3}}(U_{1}) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Mat(2, F \overline{Y} ′′_{n_{1}}) | (a + d)s = 0 = (b + c)s\} .

Obviously, I := F ∩ Rad(Mat(2, F \overline{Y} ′′_{n_{1}})) is a nilpotent ideal in F such that

F/I = F/F ∩ Rad(Mat(2, F \overline{Y} ′′_{n_{1}})) ≅ F + Rad(Mat(2, F \overline{Y} ′′_{n_{1}}))/ Rad(Mat(2, F \overline{Y} ′′_{n_{1}}))
⊆ Mat(2, F \overline{Y} ′′_{n_{1}})/ Rad(Mat(2, F \overline{Y} ′′_{n_{1}})).

Since F \overline{Y} ′′_{n_{1}} is a local F -algebra, there is an isomorphism

Mat(2, F \overline{Y} ′′_{n_{1}})/ Rad(Mat(2, F \overline{Y} ′′_{n_{1}})) ⟶ Mat(2, F).

Moreover, since the annihilator of s in F \overline{Y} ′′_{n_{1}} coincides with Rad(F \overline{Y} ′′_{n_{1}}), the above isomorphism maps the algebra F/I onto

\{(\begin{matrix}ā & b̄ \\ c̄ & d̄ \end{matrix}) ∈ Mat(2, F) | ā = d̄, b̄ = c̄\} ≅ F C_{2}.

In consequence, F/I and thus also F = End_{F \overline{Z}_{3}}(U_{1}) is a local F -algebra, and Res\begin{matrix}\overline{Y} & {′ \atop n} \\ \overline{Z} & 3 \end{matrix}(U_{1}) is indecomposable.

Similarly, φ ∈ End_{F \overline{Z}_{2}}(U_{1}) if and only if φψ′ = ψ′φ, or equivalently, if (a + d)(s + 1) = (a + d)(s + \overline{y}_{n_{1}}^{′2}) = 0 and c(s + 1) = b(s + \overline{y}_{n_{1}}^{′2}). Since both s + 1, s + \overline{y}_{n_{1}}^{′2} ∈ F \overline{Y} ′′_{n_{1}} are units, we get

F′ := End_{F \overline{Z}_{2}}(U_{1}) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Mat(2, F \overline{Y} ′′_{n_{1}}) | a = d, c = b\overline{y}_{n_{1}}^{′2}\} .

Let 0 ≠ e = a · (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) + b · (\begin{matrix}0 & 1 \\ \overline{y}_{n_{1}}^{′2} & 0 \end{matrix}) ∈ F′, for some a, b ∈ F \overline{Y} ′′_{n_{1}}, be an idempotent. Then

a · (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) + b · (\begin{matrix}0 & 1 \\ \overline{y}_{n_{1}}^{′2} & 0 \end{matrix}) = e = e^{2} = a^{2} · (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) + b^{2} · (\begin{matrix}\overline{y}_{n_{1}}^{′2} & 0 \\ 0 & \overline{y}_{n_{1}}^{′2} \end{matrix})

shows that b = 0 and a^{2} = a ≠ 0 is an idempotent in F \overline{Y} ′′_{n_{1}}. Since F \overline{Y} ′′_{n_{1}} is a local F -algebra we conclude that a = 1, thus we have e = 1, implying that F′ = End_{F \overline{Z}_{2}}(U_{1}) is a local

F -algebra as well, and Res\begin{matrix}\overline{Y} & {′ \atop n} \\ \overline{Z} & 2 \end{matrix}(U_{1}) is indecomposable.

Therefore, we have now shown that U_{1} is an indecomposable F \overline{Y} ′_{n}-module with vertex \overline{Y} ′_{n}. This in turn implies that U_{1}, as F Y′_{n}-module, is also indecomposable with vertex Y′_{n} (cf. [12]). Replacing y′_{n_{1}} by y′_{n_{2}}, we deduce that also U_{2} has vertex Y′_{n}.

Proposition 7.2. Let n_{l} = 2 and l = 3. Then E has vertex Q_{n} and source Res_{Q_{n}}^{𝔄_{n}}(E).

Proof. We first show that Res_{Q_{n}}^{𝔄_{n}}(E) is indecomposable. For this, fix an indecomposable direct sum decomposition of Res_{Q_{n}}^{𝔄_{n}}(E). For j ∈ {1, 2}, by Proposition 4.3, there is precisely one indecomposable direct summand V_{j} in this decomposition which is not annihilated by y_{n_{j}}^{′+}. We denote the preimage of V_{j} under the natural epimorphism M′ ⟶ M′/M′′ by V̂_{j}. From \overline{γ}_{j}^{+} ∈ V_{j} we, by Proposition 4.2, conclude that Soc(Res_{Y′_{n}}^{𝔄_{n}}(E)) ≤ V_{1} + V_{2}, so that Res_{Y′_{n}}^{𝔄_{n}}(E) = V_{1} + V_{2}. Hence it suffices to show that V_{1} = V_{2}.

Assume that V_{1} ≠ V_{2}. We first conclude from Lemma 7.1 that Res_{Q_{n}}^{𝔄_{n}}(E) = V_{1} ⊕ V_{2}, where dim(V_{j}) = n_{j}. As in the proof of Lemma 6.1, for j = 1, 2, 3, we use the F P_{n}-epimorphism π_{j} : M ⟶ M_{n_{j}}. We show that (π_{1})_{|V̂_{1}} is surjective: We have V̂_{2}^{π_{1}} ≤ Rad(Res_{Y′_{n_{1}}}^{P_{n}} (M_{n_{1}})), since

otherwise taking w ∈ V̂_{2} such that w^{π_{1}} ∈ M_{n_{1}} \ Rad(Res_{Y′_{n_{1}}}^{P_{n}} (M_{n_{1}})) yields w(y′_{n_{1}} + 1)^{n_{1}−1} =

aγ_{1}^{+} ∈ V̂_{2}, for some 0 ≠ a ∈ F, a contradiction. From V̂_{1} +V̂_{2} = M′ we get (V̂_{1} +V̂_{2})^{π_{1}} = M_{n_{1}}, thus V̂_{1}^{π_{1}} = M_{n_{1}}. Hence from dim(V̂_{1}) = n_{1} +1 = dim(M_{n_{1}})+1 we get dim(V̂_{1} ∩ker(π_{1})) = 1. We set v_{0} := γ_{2}^{+} + γ_{3}^{+}. Then clearly v_{0} ∈ ker(π_{1}) and, since v_{0} = γ^{+} + γ_{1}^{+}, we also have v_{0} ∈ V̂_{1}, thus V̂_{1} ∩ ker(π_{1}) = ⟨v_{0}⟩_{F}.

Now there is some w ∈ V̂_{1} such that w^{π_{1}} = γ_{1} = δ_{1}. We write w = δ_{1} + ∑_{i=n_{1}+1}^{n} a_{i}δ_{i} with appropriate a_{n_{1}+1}, …, a_{n} ∈ F. Then we have

w̃ := w(y′_{n_{2}} + 1) = ∑_{i=n_{1}+2}^{n_{1}+n_{2}}(a_{i} + a_{i−1})δ_{i} + (a_{n_{1}+n_{2}} + a_{n_{1}+1})δ_{n_{1}+1}
+ (a_{n−1} + a_{n})(δ_{n−1} + δ_{n}) ∈ V̂_{1} ∩ ker(π_{1}).

Since w ∈ M′, we have w̃ ≠ 0, hence we get w̃ = av_{0}, for some 0 ≠ a ∈ F, where we may suppose that a = 1. Thus we have a_{i+1} = a_{i} + 1 for all i ∈ {n_{1} + 1, …, n − 3} and a_{n} = a_{n−1} + 1. Hence adding a suitable multiple of v_{0} we may assume that w = δ_{1} + (b + 1)δ_{n−1} + bδ_{n} + ∑_{i=0}^{n_{2}/2−1} δ_{n_{1}+1+2i}, for some b ∈ F. Since V̂_{1} is an F Q_{n}-module, we also have

ww_{n_{2},2} = δ_{1} + (b + 1)δ_{n−1} + bδ_{n} + ∑_{i=1}^{n_{2}/2} δ_{n_{1}+2i} = w + γ_{2}^{+} ∈ V̂_{1},

implying γ_{2}^{+} ∈ V̂_{1} ∩ ker(π_{1}), a contradiction. Hence we have V_{1} = V_{2}.

It remains to show that Q_{n} is a vertex of E. We follow the strategy given in Remark 1.3, and assume that E is relatively R-projective, for some maximal subgroup R < Q_{n}. By Lemma 7.1 we have Res_{Y′_{n}}^{Q_{n}}(V_{1}) = U_{1} ⊕ U_{2} where the U_{i} are indecomposable with vertex Y′_{n}. Since Res_{Q_{n}}^{𝔄_{n}}(E) = V_{1} is indecomposable, we infer that Y′_{n} ≤ R. Thus, again by Lemma 7.1, Res_{R}^{Q_{n}}(V_{1}) is either indecomposable or the direct sum of an indecomposable module of dimension n_{1} and an indecomposable module of dimension n_{2} ≠ n_{1}, a contradiction.

8 The case n_{l} = 2 and l = 2

In order to complete the proof of Theorem 1.1, we are now left with the case where l = 2 = n_{l} which is treated in the following.

Proposition 8.1. Let n_{l} = 2 and l = 2. If n > 6 then E has vertex Q_{n} and source Res_{Q_{n}}^{𝔄_{n}}(E).

Proof. We follow the strategy given in Remark 1.3, and assume that E is relatively Rprojective for some maximal subgroup R < Q_{n}. Hence we have ⟨Φ(Q_{n}), Q_{n−4}⟩ ≤ R and, in particular, X_{n_{1}} = X_{n} ≤ Φ(Q_{n}) ≤ R. Using Φ(Q_{n}) = Φ(P_{n}) = Φ(P_{n_{1}}) and Q_{\frac{n_{1}}{2}} ≤ Q_{n−4}, Proposition 2.6 yields B′_{n_{1}} = Q_{\frac{n_{1}}{2}} Φ(P_{n_{1}}) ≤ Q_{n−4}Φ(Q_{n}) ≤ R. Since Q_{n}/B′_{n_{1}} is elementary abelian of order 4, there are precisely three maximal subgroups of Q_{n} containing B′_{n_{1}} = B′_{n}:

R′_{1} := B′_{n}⟨y′_{n_{1}}⟩, R′_{2} := B′_{n}⟨w_{n_{1},1}⟩ = Q_{n_{1}}, R′_{3} := B′_{n}⟨w_{2,1}w_{2,2}⟩ = (B_{n} × P_{2}) ∩ Q_{n}.

We next show that E restricts indecomposably to R′_{1} and R′_{2}: By Remark 4.4, part (a), and using dim(E) = n_{1} = 2 dim(F X_{n_{1}}) we get Res_{X_{n_{1}}}^{𝔄_{n}} (E) ≅ F X_{n_{1}} ⊕ F X_{n_{1}}. Now let i ∈ {1, 2},

fix an indecomposable direct sum decomposition of Res_{R′_{i}}^{𝔄_{n}}(E), and assume that Res_{R′_{i}}^{𝔄_{n}}(E) is

decomposable. Hence Res_{R′_{i}}^{𝔄_{n}}(E) actually has two indecomposable summands, both of which are not annihilated by x_{n_{1}}^{+}. But, since R_{i}^{′p_{1}q_{1}} = ⟨w_{n_{1},1}⟩, Remark 4.4, part (b), implies that there is precisely one such summand, a contradiction. Hence from this we also conclude that Res_{Q_{n}}^{𝔄_{n}}(E) is indecomposable.

It remains to show that E is not relatively R′_{3}-projective. We claim that it suffices to show that E restricts indecomposably to H′ := (𝔖_{\frac{n_{1}}{2}} × 𝔖_{\frac{n_{1}}{2}} × 𝔖_{2}) ∩ 𝔄_{n}. Namely, then E also

restricts indecomposably to Ĥ′ := ((𝔖_{\frac{n_{1}}{2}} ≀ 𝔖_{2}) × 𝔖_{2}) ∩ 𝔄_{n}. Of course E is relatively Ĥ′-

projective, hence E and Res𝔄_{n} ; Ĥ′(E) have a common vertex. If E were relatively R′_{3}-projective then Res𝔄_{n} ; Ĥ′(E) would be relatively H′-projective, a contradiction since |Ĥ′ : H′| = 2.

Therefore, we now show that End_{F H′}(E) is a local F -algebra so that Res_{H′}^{𝔄_{n}}(E) is indecomposable. Using the F -basis \{\overline{γ}_{1} + \overline{γ}_{n}, …, \overline{γ}_{n_{1}} + \overline{γ}_{n}\} of D, we get

Res_{𝔖_{\frac{n_{1}}{2}} ×𝔖_{\frac{n_{1}}{2}}}^{𝔖_{n}} (D) = M_{1} ⊕ M_{2} ≅ (M^{( \frac{n_{1}}{2} −1,1)} ⊠ F) ⊕ (F ⊠ M^{( \frac{n_{1}}{2} −1,1)}).

Here M^{( \frac{n_{1}}{2} −1,1)} ⊠ F denotes the outer tensor product of the F 𝔖_{\frac{n_{1}}{2}} -modules M^{( \frac{n_{1}}{2} −1,1)} and F. Furthermore, M_{1} has F -basis \{\overline{γ}_{1} + \overline{γ}_{n}, …, \overline{γ}_{\frac{n_{1}}{2}} + \overline{γ}_{n}\}, and M_{2} has F -basis \{\overline{γ}_{\frac{n_{1}}{2} +1} + \overline{γ}_{n}, …, \overline{γ}_{n_{1}} + \overline{γ}_{n}\}. Both modules are indecomposable and uniserial with descending composition factors

(F, D^{( \frac{n_{1}}{2} −1,1)} ⊠ F, F) and (F, F ⊠ D^{( \frac{n_{1}}{2} −1,1)}, F),

respectively. Note that this also holds for n = 6, that is n_{1} = 4, if we just let D^{(1^{2})} := {0}. Now, since n > 6, we have n_{1} ≥ 8. Consequently Res_{𝔄_{\frac{n_{1}}{2}} ×𝔄_{\frac{n_{1}}{2}}}^{𝔖_{n}} (D) = M̃_{1} ⊕ M̃_{2}, where M̃_{1} and

M̃_{2}, for n_{1} > 8, are uniserial with descending composition factors

(F, E_{0}^{( \frac{n_{1}}{2} −1,1)} ⊠ F, F) and (F, F ⊠ E_{0}^{( \frac{n_{1}}{2} −1,1)}, F),

respectively. For n_{1} = 8 we have Soc(M̃_{1}) ≅ M̃_{1}/ Rad(M̃_{1}) ≅ F and Rad(M̃_{1})/ Soc(M̃_{1}) ≅ (E_{+}^{(3,1)} ⊕ E_{−}^{(3,1)}) ⊠ F as well as Soc(M̃_{2}) ≅ M̃_{2}/ Rad(M̃_{2}) ≅ F and Rad(M̃_{2})/ Soc(M̃_{2}) ≅ F ⊠ (E_{+}^{(3,1)} ⊕ E_{−}^{(3,1)}). Thus

E := End_{F [𝔖_{\frac{n_{1}}{2}} ×𝔖_{\frac{n_{1}}{2}} ]}(D) = End_{F [𝔄_{\frac{n_{1}}{2}} ×𝔄_{\frac{n_{1}}{2}} ]}(D)

has dimension 6, and F -basis {φ_{1}, …, φ_{6}} where φ_{1} and φ_{2} are the projections onto M̃_{1} and M̃_{2}, respectively, φ_{3} annihilates M̃_{2} and maps M̃_{1} onto Soc(M̃_{1}), φ_{4} annihilates M̃_{1} and maps M̃_{2} onto Soc(M̃_{2}), φ_{5} annihilates M̃_{2} and maps M̃_{1} onto Soc(M̃_{2}), φ_{6} annihilates M̃_{1} and maps M̃_{2} onto Soc(M̃_{1}). That is, for a_{1}, …, a_{n_{1}} ∈ F and v := ∑_{i=1}^{n_{1}} a_{i}(\overline{γ}_{i} + \overline{γ}_{n}) ∈ D, we may suppose that

v^{φ_{1}} = a_{1}(\overline{γ}_{1} + \overline{γ}_{n}) + ⋯ + a_{\frac{n_{1}}{2}} (\overline{γ}_{\frac{n_{1}}{2}} + \overline{γ}_{n}),
v^{φ_{2}} = a_{\frac{n_{1}}{2} +1}(\overline{γ}_{\frac{n_{1}}{2} +1} + \overline{γ}_{n}) + ⋯ + a_{n_{1}}(\overline{γ}_{n_{1}} + \overline{γ}_{n}),
v^{φ_{3}} = (a_{1} + ⋯ + a_{\frac{n_{1}}{2}} )((\overline{γ}_{1} + \overline{γ}_{n}) + ⋯ + (\overline{γ}_{\frac{n_{1}}{2}} + \overline{γ}_{n})),
v^{φ_{4}} = (a_{\frac{n_{1}}{2} +1} + ⋯ + a_{n_{1}})((\overline{γ}_{\frac{n_{1}}{2} +1} + \overline{γ}_{n}) + ⋯ + (\overline{γ}_{n_{1}} + \overline{γ}_{n})),
v^{φ_{5}} = (a_{1} + ⋯ + a_{\frac{n_{1}}{2}} )((\overline{γ}_{\frac{n_{1}}{2} +1} + \overline{γ}_{n}) + ⋯ + (\overline{γ}_{n_{1}} + \overline{γ}_{n})),
v^{φ_{6}} = (a_{\frac{n_{1}}{2} +1} + ⋯ + a_{n_{1}})((\overline{γ}_{1} + \overline{γ}_{n}) + ⋯ + (\overline{γ}_{\frac{n_{1}}{2}} + \overline{γ}_{n})).

The multiplication in E is given by

φ_{1}^{2} = φ_{1},  φ_{1}φ_{3} = φ_{3} = φ_{3}φ_{1},  φ_{6}φ_{1} = φ_{6} = φ_{2}φ_{6},
φ_{2}^{2} = φ_{2},  φ_{2}φ_{4} = φ_{4} = φ_{4}φ_{2},  φ_{5}φ_{2} = φ_{5} = φ_{1}φ_{5},

and any other product of two basis elements vanishes. Note that H := 𝔖_{\frac{n_{1}}{2}} × 𝔖_{\frac{n_{1}}{2}} × 𝔖_{2} = (𝔖_{\frac{n_{1}}{2}} × 𝔖_{\frac{n_{1}}{2}} )⟨(n − 1, n)⟩ and H′ = (𝔄_{\frac{n_{1}}{2}} × 𝔄_{\frac{n_{1}}{2}} )⟨(1, 2)(\frac{n_{1}}{2} + 1, \frac{n_{1}}{2} + 2), (1, 2)(n − 1, n)⟩. The algebra E carries an H-algebra structure with respect to the conjugation action. We thus deduce that Ẽ := End_{F H′}(E) = End_{F H}(D), and Ẽ consists of those φ ∈ E which are fixed under (n − 1, n). Let now v ∈ D be as above, and let φ := b_{1}φ_{1} + ⋯ + b_{6}φ_{6} ∈ E for some b_{1}, …, b_{6} ∈ F. Then

v(n − 1, n) = ∑_{i=1}^{n_{1}}(∑_{j≠i} a_{j})(\overline{γ}_{i} + \overline{γ}_{n}) = v^{φ_{1}+···+φ_{6}}.

Hence φ ∈ Ẽ if and only if φ(φ_{1} + ⋯ + φ_{6}) = (φ_{1} + ⋯ + φ_{6})φ, or equivalently, if b_{1} = b_{2}. We have thus shown that Ẽ has dimension 5 and F -basis {φ_{1} + φ_{2}, φ_{3}, φ_{4}, φ_{5}, φ_{6}}. Since Ẽ is abelian and φ_{3}, φ_{4}, φ_{5}, φ_{6} are nilpotent, we also deduce that Rad(Ẽ) has dimension 4. In particular, Ẽ is local, and the assertion follows.

Remark 8.2. It remains to consider the case n = 6. Let E := E_{0}^{(5,1)}. In view of the above observations, we aim to show that E has vertex Q := (𝔖_{2} × 𝔖_{2} × 𝔖_{2}) ∩ 𝔄_{6} = ⟨(1, 2)(3, 4), (3, 4)(5, 6)⟩ ≅ V_{4} and sources of dimension 2: To this end, let Q < Q_{6} := ⟨(1, 3)(2, 4), (1, 2)(3, 4), (3, 4)(5, 6)⟩ < 𝔄_{6}, where Q_{6} is a dihedral group of order 8. With respect to the basis {γ̄_{1} + γ̄_{6}, γ̄_{2} + γ̄_{6}, γ̄_{3} + γ̄_{6}, γ̄_{4} + γ̄_{6}}, the action of Q_{6} on E is given via the following matrices:

(1, 3)(2, 4) ⟷ (\begin{matrix}. & . & 1 & . \\ . & . & . & 1 \\ 1 & . & . & . \\ . & 1 & . & . \end{matrix}) ,

(1, 2)(3, 4) ⟷ (\begin{matrix}. & 1 & . & . \\ 1 & . & . & . \\ . & . & . & 1 \\ . & . & 1 & . \end{matrix}) , (3, 4)(5, 6) ⟷ (\begin{matrix}. & 1 & 1 & 1 \\ 1 & . & 1 & 1 \\ 1 & 1 & 1 & . \\ 1 & 1 & . & 1 \end{matrix}) .

Let ω ∈ F be a primitive third root of unity, and consider the following F -subspaces of E:

U := ⟨(γ̄_{1} + γ̄_{6}) + ω(γ̄_{4} + γ̄_{6}), (γ̄_{2} + γ̄_{6}) + ω(γ̄_{3} + γ̄_{6})⟩_{F} ,
V := ⟨(γ̄_{3} + γ̄_{6}) + ω(γ̄_{2} + γ̄_{6}), (γ̄_{4} + γ̄_{6}) + ω(γ̄_{1} + γ̄_{6})⟩_{F} .

Then we have E = U ⊕ V as F -vector spaces, and the action of Q_{6} on E, with respect to this new F -basis, is given via:

(1, 3)(2, 4) ⟷ (\begin{matrix}. & . & 1 & . \\ . & . & . & 1 \\ 1 & . & . & . \\ . & 1 & . & . \end{matrix}) ,

(1, 2)(3, 4) ⟷ (\begin{matrix}. & 1 & . & . \\ 1 & . & . & . \\ . & . & . & 1 \\ . & . & 1 & . \end{matrix}) , (3, 4)(5, 6) ⟷ (\begin{matrix}ω & ω^{2} & . & . \\ ω^{2} & ω & . & . \\ . & . & ω^{2} & ω \\ . & . & ω & ω^{2} \end{matrix}) .

Thus both U and V are F Q-submodules of E, and we have Res_{Q}^{𝔄_{6}}(E) = U ⊕V, where the socle of U has dimension 1 so that U is indecomposable. Moreover, from ((1, 2)(3, 4))^{(1,3)(2,4)} = (1, 2)(3, 4) and

((3, 4)(5, 6))^{(1,3)(2,4)} = (1, 2)(5, 6) = (1, 2)(3, 4) · (3, 4)(5, 6)

we infer that Ind_{Q}^{Q_{6}}(U) ≅ Res_{Q_{6}}^{𝔄_{6}}(E). Thus, by Green’s Indecomposability Theorem, Res_{Q_{6}}^{𝔄_{6}}(E) is indecomposable, and E is relatively Q-projective. Moreover, each proper subgroup of Q is cyclic, and since E belongs to a block of F 𝔄_{6} with non-cyclic defect groups, E cannot have a cyclic vertex, by [5]. So Q and U then have to be vertex and source, respectively, of E.

9 Appendix

We give a new corrected proof of [13, Thm. 1.4(a), 1.5(a)]. For the case l = 2 and n_{l} = 2 we reuse the observations made in the proof of Proposition 8.1, where we actually have already chosen notation appropriately.

Theorem 9.1. Let n be even, but not a 2-power. Then D has vertex P_{n} and source Res_{P_{n}}^{𝔖_{n}}(D).

Proof. By [13, Prop. 3.7], Res_{P_{n}}^{𝔖_{n}}(D) is indecomposable. We again follow the strategy given in Remark 1.3. Assume that R < P_{n} is a maximal subgroup such that D is relatively R-projective, and fix an indecomposable direct sum decomposition of Res_{R}^{𝔖_{n}}(D). We have X_{n} ≤ Φ(P_{n}) ≤ R. Moreover, we have Res_{𝔖_{n−1}}^{𝔖_{n}}(D) ≅ D^{(n−2,1)}. The latter, by [13, Thm. 1.2, 1.3], has vertex P_{n−4} and trivial source. Thus we have P_{n−4} ≤ R.

Let first n_{l} > 2, or n_{l} = 2 and l ≥ 3; recall that l ≥ 2 anyway. Then, in all these cases we have P_{n_{1}} ≤ P_{n−4} ≤ R, implying that R^{p_{1}q_{1}} = ⟨w_{n_{1},1}⟩. Then by Remark 4.4, part (b), there is precisely one indecomposable direct summand U in the decomposition which is not annihilated by x_{n_{1}}^{+}. Thus (F X_{n_{1}} ⊕ F X_{n_{1}})| Res_{X_{n_{1}}}^{R} (U) implies dim(U) > dim(D)/2, a contradiction.

Let now n_{l} = 2 and l = 2. Then using Φ(P_{n}) = Φ(P_{n_{1}}) and P_{\frac{n_{1}}{2}} ≤ P_{n−4}, Proposition 2.6 yields B_{n} = B_{n_{1}} = P_{\frac{n_{1}}{2}} Φ(P_{n_{1}}) ≤ P_{n−4}Φ(P_{n}) ≤ R. Since P_{n}/B_{n} is elementary abelian of order 4, there are precisely three maximal subgroups of P_{n} containing B_{n}, these are R_{1} := B_{n}⟨y′_{n_{1}}⟩ and R_{2} := B_{n}⟨y_{n_{1}}⟩ = P_{n_{1}} and R_{3} := B_{n}⟨w_{2,2}⟩ = B_{n} × P_{2}. With the notation as in the proof of Proposition 8.1, we have R_{i} ∩𝔄_{n} = R′_{i} for i ∈ {1, 2, 3}. Hence from the proof of Proposition 8.1 we infer that D restricts indecomposably to R_{1} and R_{2}.

It remains to show that D is not relatively R_{3}-projective. As in the proof of Proposition 8.1, it suffices to show that D restricts indecomposably to H := 𝔖_{\frac{n_{1}}{2}} × 𝔖_{\frac{n_{1}}{2}} × 𝔖_{2}. Namely,

then D also restricts indecomposably to Ĥ := (𝔖_{\frac{n_{1}}{2}} ≀ 𝔖_{2}) × 𝔖_{2}. Of course D is relatively Ĥ-

projective, hence D and Res𝔖_{n} ; Ĥ(D) have a common vertex. If D were relatively R_{3}-projective then Res𝔖_{n} ; Ĥ(D) would be relatively H-projective, a contradiction since |Ĥ : H| = 2. But now the proof of Proposition 8.1 shows that Ẽ = End_{F H}(D) is a local F -algebra so that Res_{H}^{𝔖_{n}}(D) indeed is indecomposable.

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S.D.: Mathematical Institute, University of Oxford 24-29 St Giles’, Oxford, OX1 3LB, UK danz@maths.ox.ac.uk

J.M.: Lehrstuhl D für Mathematik, RWTH Aachen Templergraben 64, D-52062 Aachen, Germany Juergen.Mueller@math.rwth-aachen.de