PARAMETRIZATION OF CENTRAL FRATTINI EXTENSIONS AND ISOMORPHISMS OF SMALL GROUP RINGS

MARTIN HERTWECK AND MARCOS SORIANO

Abstract. We solve the modular isomorphism problem for small group rings, i.e., we determine, for a given finite p-group H, precisely which central Frattini extensions of H give rise to isomorphic small group rings over the field with p elements.

1. Introduction

Let G be a finite p-group, where p is an arbitrary rational prime. Any descending central series

G = G_{1} ≥ G_{2} ≥ … ≥ G_{l} ⪈ G_{l+1} = 1

with G_{2} = Φ(G) = [G, G]G^{p}, the Frattini subgroup of G, and elementary abelian sub-quotients G_{i}/G_{i+1}, allows us to describe G as being successively built from extensions 𝔼_{1}, …, 𝔼_{l−1}:

[figure: 𝔼_{i} :; 1; G_{i+1}/G_{i+2}; G/G_{i+2}; G/G_{i+1}; 1.]

Note that, by assumption, G_{i+1}/G_{i+2} is a central elementary abelian subgroup contained in the Frattini subgroup of G/G_{i+2}, which shall be expressed shortly by saying that 𝔼_{i} is a central Frattini extension.

An important example of such a central series is the Brauer–Jennings–Zassenhaus series G = M_{1}(G) ≥ M_{2}(G) ≥ … (see [13] and Section 3). It appears naturally in the context of modular group algebras, since M_{n}(G) is the so-called n-th dimension subgroup of G:

M_{n}(G) = G ∩ (1 + I(kG)^{n}),

where I(kG) denotes the augmentation ideal of the modular group ring kG, with k = 𝔽_{p} being the field with p elements.

A group basis of kG is a subgroup of the group of units of kG which constitutes a basis of the vector space kG over k. The Modular Isomorphism Problem (usually abbreviated as MIP) asks whether, for a given p-group G, a group basis of kG must be necessarily isomorphic to G.

The results obtained by Passi and Sehgal in [14] and Ritter and Sehgal in [16] say — to be read as successive improvements — that the isomorphism classes of the sub-quotients

M_{n}(G̃)/M_{n+1}(G̃), M_{n}(G̃)/M_{n+2}(G̃), M_{n}(G̃)/M_{2n+1}(G̃)

are independent of the chosen group basis G̃ of kG.

Date: March 2, 2005.

2000 Mathematics Subject Classification. Primary 20D15; Secondary 20E22.

The first author acknowledges support by the Deutsche Forschungsgemeinschaft.

Keywords and Phrases. p-group, central extension, small group ring.

They seem to suggest that an inductive approach to (MIP) should be possible, a point of view taken in the recent attack of Borge and Laudal on (MIP) (see [2,3], as well as the discussion of this approach given by the authors in [8, §3]). Trying to substantiate the attempt of [2,3] was the starting point of the investigation that led to the present paper.

It appears natural to consider for a fixed p-group H the class 𝔽(H) of all short exact sequences of groups

[figure: 𝔼 :; 1; V; E; H; 1,]

where V is a central elementary abelian p-subgroup contained in the Frattini subgroup of E (see Definition 4.1). The small group ring of E with respect to V is the quotient

s(E, V) = \frac{kE}{I(kV)I(kE)}.

Associated to 𝔼 there is a short exact sequence of k-algebras, the sequence for the small group ring (see Section 3):

[figure: 𝕊 :; 0; {V − 1}; s(E, V); kH; 0.]

We introduce two equivalence relations on 𝔽(H) — isomorphism and s-equivalence —, which essentially describe isomorphisms between group sequences 𝔼 or, rather, small group ring sequences 𝕊 (see Definition 4.3).

The main result of the paper, Theorem 4.4, describes the s-equivalence classes of extensions in 𝔽(H) in terms of an action of the outer automorphism group Out(kH) on the kernel V of a certain ‘universal’ extension 𝔼, given in (4.3). There is a natural inclusion Out(H) ⊆ Out(kH), and restriction of the action to Out(H) yields precisely the isomorphism classes of extensions in 𝔽(H). This solves, in view of Corollary 3.7, the isomorphism problem for small group rings in an optimal way.

With the guide of Theorem 4.4, it is fairly easy to verify that for the dihedral group D_{8} of order 8, the (non-isomorphic) central Frattini extensions in 𝔽(D_{8}) with middle terms isomorphic to the dihedral, semi-dihedral and generalized quaternion group of order 16 give rise to isomorphic small group rings, and to construct two pairs of central Frattini extensions in 𝔽(D_{8}) with non-isomorphic middle groups of order 32 which give rise to isomorphic small group rings (see Example 5.3). Although announced in the literature (see the (foot)note (e) contained in the remark following Theorem 6.25 in [21]), such examples do not seem to have been published until now.

For the orientation of the reader, we briefly describe the contents of each section. Section 2 contains the fairly elementary group-theoretical Lemma 2.1 on the

lifting of group homomorphisms in a special situation, which, as a consequence, allows the — completely cohomology-free — introduction of the key concept of obstruction space (see Proposition 2.2). The reader interested in an alternative approach to this concept may wish to consult [8].

Besides giving a short introduction into small group rings, the purpose of Section 3 is twofold: first, to transfer the concept of obstruction spaces to the setting of small group rings (see Corollary 3.2), and second, to show that s-equivalence of central Frattini extensions as introduced in Section 4 (see Definition 4.3) is nothing but isomorphism of small group rings (see Corollary 3.7).

In Section 4 we introduce central Frattini extensions (see Definition 4.1) and study the class 𝔽(H) of all such extensions with fixed finite factor p-group H with respect to two closely related equivalence relations: isomorphism and s-equivalence (see Definition 4.3). A key observation (see Lemma 4.2) relates obstruction spaces to sequences in 𝔽(H). The main Theorem 4.4 may be interpreted as the construction of all classes of sequences in 𝔽(H) as appropriate quotients of a certain ‘universal’ central Frattini extension (4.3). Thereby, the action of the groups Out(H) and Out(kH) on the subspaces U of the kernel V of this extension, which is described in detail in Section 4, identifies isomorphic respectively s-equivalent sequences by interpreting the subspaces U as obstruction spaces in the group respectively small group ring setting.

It might be helpful to complement the reading of Sections 3 and 4 with looking at the examples from Section 5. Example 5.1 contains all of what can be said about the simple (and exceptional) case of cyclic groups, giving also an easy, explicit example of small group rings. Example 5.2 aims at describing 𝔽(H) for H elementary abelian. It turns out that in this case isomorphism and s-equivalence classes coincide, and their explicit description is reduced to a combinatorial problem related to certain GL(𝔽_{p})-modules. Finally, due to a ‘strictly larger’ action of Out(𝔽_{2}D_{8}) when compared to the action of Out(D_{8}) (D_{8} denoting the dihedral group of order 8), we are able to construct in Example 5.3 non-isomorphic groups having isomorphic modular small group rings with quotient algebra 𝔽_{2}D_{8}. The reader is invited to extend this family of examples by working with GAP [7], our favourite computer algebra system in this context.

The final Section 6 is mainly devoted to the formulation of a new proof of the result of Röhl [20] in our setting (see Theorem 6.8), thus relating it to small group rings. Roughly speaking, the theorem is about the modular isomorphism problem for p-groups allowing a certain type of presentation. En passant, we derive a couple of (hopefully) useful results, from which Lemma 6.7 and Corollary 6.9 might be highlighted.

2. A lemma about lifting of group homomorphisms

Let G be a finite group. We shall consider short exact sequences 𝔼 of finite groups together with homomorphisms φ from G to the end term of the sequence:

(2.1)  [figure: 𝔼_{φ} :; 1; N; E; E/N; 1; G; φ]

We shall refer to 𝔼_{φ} just as a diagram. We will say that φ lifts modulo M, where M is a normal subgroup of E contained in N, if there exists a homomorphism φ̂ : G E/M such that the following diagram, where π : E E/M is the natural map, is commutative:

(2.2)  [figure: E/M; E/N; 1; E; π; φ̂; E/N; ‖; 1; G; φ]

Then, the homomorphism φ̂ will be called a lifting of φ.

Sometimes it is possible to make a statement about the collection of all normal subgroups modulo which φ lifts. We show:

Lemma 2.1. Suppose that a diagram 𝔼_{φ} is given, as shown in (2.1), and that N is a direct product of minimal normal subgroups of E. Then there exists a unique normal subgroup U of E contained in N which is minimal in the following sense: φ can be lifted modulo U, and any other normal subgroup modulo which φ lifts contains U.

Proof. We begin with a preliminary remark about the normal subgroups of E which are contained in N: If M is such a subgroup, M ≠ 1, and N = N_{1} × … × N_{r} with minimal normal subgroups N_{i} of E, then we have, possibly after a renumbering of the N_{i}, that N = M × N_{1} × … × N_{s} for some s < r. This can be proved by induction on the order of M as follows. First of all, the projection of M onto some N_{i_{0}} is surjective. Let L be the kernel of this surjection. If L = 1, then N is the direct product of M and the N_{i}, i ≠ i_{0}. Otherwise L ≠ 1, and we may assume by induction on |M| that N = L × K with K = N_{1} × … × N_{s+1}, some s < r − 1, after a suitable renumbering. Note that M = L × (M ∩ K) and that M ∩ K is a minimal normal subgroup of E contained in K. As before (the case L = 1), we have K = (M ∩ K) × N_{1} × … × N_{s} after renumbering, which proves the claim. In particular, any normal subgroup M of E contained in N has a direct complement in N which is normal in E.

Now let U be a normal subgroup of E contained in N such that φ lifts modulo U, but not modulo any normal subgroup which is properly contained in U (possibly U = N). By way of contradiction, suppose that V is a normal subgroup of E contained in N such that φ lifts modulo V, but that V does not contain U.

We assume without lost of generality that the injection N E is in fact inclusion.

If U ∩ V ≠ 1, we may consider the diagram

[figure: 1; N/U ∩ V; E/U ∩ V; E/N; 1.; G; φ]

Then φ lifts modulo U/U ∩ V and modulo V/U ∩ V. By induction on the order of E, we can assume that φ lifts modulo the intersection of these subgroups, i.e., that there exists a lifting φ̂: G E/U ∩ V of φ, contradicting our choice of U since U ∩ V is a proper subgroup of U.

Hence we can assume that U ∩ V = 1. Then there exists a normal subgroup W of E such that N = U × V × W. Let α : G E/U and β : G E/V be liftings of φ. Let σ be the composite of α and the natural map E/U E/UV, and let τ be the composite of β and the natural map E/V E/UV. If π : E/UV E/N is the natural map, then σπ = τπ since both σπ and τπ are the composite of φ and the inverse of the automorphism of E/N which the given surjection E E/N induces. At last, let β̄ denote the composite of β and the natural map E/V E/V W. Now consider the following commutative diagram, in which each square is a pullback diagram where the maps are the natural ones. Note that pullback diagrams have a universal mapping property (as described in [6, §3, Exercise 11] for pullback diagrams of modules).

[figure: G; E; φ̂; E/V; E/V W; E/U; E/UV; E/N; E/UW; E/N; π; π; σ; τ; ψ; β̄; α]

As σπ = τπ, we obtain a homomorphism ψ : G E/V such that the composite of ψ with the natural map E/V E/V W is β̄, and the composite of ψ with the natural map E/V E/UV is σ. It follows that α and ψ induce a homomorphism φ̂ : G E, which is a lifting of φ. Consequently U = 1, contradicting our assumption that V does not contain U, and the lemma is proved. □

The hypothesis of Lemma 2.1 is naturally satisfied if N is an elementary abelian p-group (for some prime p) which is contained in the center of E. In this case, we shall write obspace 𝔼_{φ} for the subgroup U described in Lemma 2.1 as we think of the vector space U as an “obstruction space.” As we shall deal only with such extensions, we record this conclusion in a separate proposition:

Proposition 2.2 (and Definition). Let 𝔼_{φ} be a diagram as shown below, where V is an elementary abelian p-group contained in the center of E (and V ^{⊂} E denotes inclusion). Then among the subgroups U of V which admit a lifting, as shown below, there is a unique smallest one, called obspace 𝔼_{φ}.

[figure: 1; V/U; ⊂; E/U; H; 1]

[figure: 𝔼_{φ} :; 1; V; ⊂; E; φ̂; H; ‖; 1]

G^{φ }

The concept of obstruction spaces for diagrams of p-groups was recently picked up by Borge and Laudal in their attack on the modular isomorphism problem (see [3]). In [8], the authors presented an elementary cohomological argument giving obspace 𝔼_{φ} as the image of a certain homomorphism which is naturally assigned to a 2-cocycle associated with 𝔼_{φ}, closely following the path marked in [3, Section 2].

3. A glance on algebras: The small group ring

The group-theoretical Proposition 2.2 has an analogue on the algebra-side. We do not go into details but present some ideas by means of the “small group ring”.

Originally, the small group ring was introduced to deal with problems for integral group rings, leading to Whitcomb’s result [23]. See [18, 1.1.8] for more details.

Let E be a finite group which has, for some prime p, an elementary abelian normal p-subgroup V. Set k = 𝔽_{p}, the field with p elements. We write I(kE) for the augmentation ideal of kE (the elements of kE with sum-of-coefficients equal to zero). Note that I(kV)kE is the kernel of the natural homomorphism kE kE/V. There is an isomorphism of abelian groups

V ≅ \frac{I(kV)kE}{I(kV)I(kE)} (3.1)

sending v ∈ V to the coset of v − 1. Indeed, this is a surjective homomorphism since vw − 1 = (v − 1) + (w − 1) + (v − 1)(w − 1) for all v, w ∈ V. Note that I(kV)kE has dimension |E| − |E/V |, so if T is a system of coset representatives to V in E then {(v − 1)t | 1 ≠ v ∈ V, t ∈ T } is a k-basis of I(kV)kE and we obtain a homomorphism I(kV)kE V of vector spaces by sending (v − 1)t to v. Since the kernel of this homomorphism contains I(kV)I(kE), (3.1) follows.

For U ≤ V, we suggestively write {U − 1} for the image of U under the isomorphism (3.1). The small group ring of E over k associated with V is the quotient

s(E, V) = \frac{kE}{I(kV)I(kE)}.

Thus we have an exact sequence

[figure: 𝕊 :; 0; {V − 1}; s(E, V); kE/V; 0.]

Since {V − 1} is an ideal of square zero, we also have an exact sequence of multiplicative groups of units:

[figure: 1; V; s(E, V)^{×}; (kE/V)^{×}; 1.]

Note that the pre-image of E/V in s(E, V)^{×} is precisely E.

Let U ≤ V. Then U is a normal subgroup of E if and only if {U − 1} is an ideal of s(E, V). Furthermore, if one of these conditions hold, we have a commutative diagram, in which the vertical maps are the natural isomorphisms:

[figure: 0; \{V/U − 1\}; s(E/U, V/U); kE/V; 0; 0; \frac{\{V − 1\}}{\{U − 1\}}; \frac{s(E, V)}{\{U − 1\}}; kE/V; ‖; 0] (3.2)

Set H = E/V. For a finite group G, consider the exact sequence 𝕊 for the small group ring together with a homomorphism φ from kG to the end term of the sequence:

[figure: 𝕊_{φ} :; 0; {V − 1}; s(E, V); κ; kH; 0; kG; φ]

Assume that a group basis G̃ of kG is mapped under φ to a group basis H̃ of kH, and let Ẽ be the pre-image κ^{−1}(H̃) of H̃ in s(E, V)^{×}. Switching to units, we get a diagram for groups:

[figure: 𝔼̃_{φ} :; 1; V; Ẽ; κ; H̃; 1; G̃; φ]

In the present case, the following simple lemma describes the transformation between lifting group and algebra homomorphisms.

Lemma 3.1. With the notation as above, let U be a normal subgroup of E contained in V. Then the following are equivalent:

(i) There exists an algebra homomorphism α making the following diagram commutative:

[figure: 0; \frac{\{V − 1\}}{\{U − 1\}}; \frac{s(E, V)}{\{U − 1\}}; kH; 0; kG; φ; α]

(ii) There exists a group homomorphism β making the following diagram commutative:

[figure: 1; V/U; Ẽ/U; H̃; 1; G̃; φ; β]

Proof. Note that Ẽ/U embeds naturally into the group of units of s(E, V)/{U −1}.

(i)⟹(ii): Let Q be the pre-image of G̃α in s(E, V). By commutativity, Qκ = H̃, so Q ≤ Ẽ, and we obtain β as a composite map, as shown in the left diagram below.

[figure: Ẽ/U; H̃; im α; β; H̃; ‖; G̃; φ; α; \frac{s(E, V)}{\{U − 1\}}; kH; kẼ/U; α; kH̃; ‖; kG; φ; β]

(ii)⟹(i): Since kG = kG̃ and kH = kH̃, we obtain α as a composite map, as shown in the right diagram above. □

From Proposition 2.2 and Lemma 3.1 we immediately obtain the following two corollaries.

Corollary 3.2. Suppose that V is central in E. Among the ideals I of s(E, V) contained in {V − 1} for which there exists an algebra homomorphism α making the following diagram commutative:

[figure: 0; {V − 1}/I; s(E, V)/I; kH; 0,; kG; φ; α]

there exists a unique minimal one (with respect to inclusion), namely I = {U − 1} with U = obspace 𝔼̃_{φ}. □

Corollary 3.3. Suppose that V is central in E, and that Gφ = H, so that we have another diagram

[figure: 𝔼_{φ} :; 1; V; E; H; 1; G; φ]

Then obspace 𝔼_{φ} = obspace 𝔼̃_{φ}. □

Though the groups G and G̃ (a priori) might be assumed to be non-isomorphic this should not come as a surprise since we may very well have E ≠ Ẽ.

In the remaining part of this section we consider isomorphisms between small group rings s(G, A) and s(H, B) of p-groups G and H associated with central elementary abelian subgroups A and B, and address the question whether {A − 1} must be mapped to {B − 1}. Apart from some minor exceptions, the answer will be in the affirmative. We begin with:

Lemma 3.4. Let G be a finite p-group, and let A be a central elementary abelian subgroup of G with 1 ≠ A < G. Let T be a set of coset representatives of A in G, and let τ be the image of ∑_{t∈T} t in s(G, A). Then the annihilator of the radical of s(G, A) is {A − 1} + kτ unless G contains an element of order p|G/A|, when the annihilator is {A − 1} (note that in the latter case, G/A is cyclic).

Proof. Let π : kG s(G, A) be the natural map, let m ∈ kG, and suppose that mπ annihilates the radical I(kG)π of s(G, A). Write m = ∑_{t∈T} m_{t}t with all m_{t} in kA. We have ∑_{t} m_{t}t ≡ ∑_{t} ε(m_{t})(t − 1) + ∑_{t} m_{t} mod I(kA)I(kG), where ε denotes the augmentation map. Since ε(m) = 0 (otherwise m would be a unit), (∑_{t} m_{t})π ∈ {A − 1}. Note that {A − 1} annihilates the radical. Set m̃ = ∑_{t} ε(m_{t})(t − 1). Then m̃π annihilates the radical, so if κ : kG kG/A denotes the natural map, then m̃κ annihilates I(kG/A). It follows that m̃κ ∈ k(∑_{x∈G/A} x), and m̃π ∈ {A − 1} + kτ where τ = (∑_{t∈T} t)π as above. It remains to check when τ annihilates the radical. Let g ∈ G. Then (∑_{t} t)g = ∑_{t} a_{t}t with all a_{t} ∈ A. Using the formula xy −1 = (x−1)+(y −1)+(x−1)(y −1), we obtain modulo I(kA)I(kG)

∑_{t} a_{t}t = ∑_{t}(a_{t}t − 1) ≡ ∑_{t}(a_{t} − 1) + ∑_{t}(t − 1) ≡ ( ∏_{t} a_{t}) − 1 + ∑_{t} t. (3.3)

The product ∏_{t} a_{t} is the image of g under the transfer homomorphism G A (see [17, 10.1]), and it follows that ∏_{t} a_{t} ≠ 1, i.e., τ(gπ) ≠ τ, if and only if g has order p|G/A| (cf. [17, 10.1.2]), and the lemma is proved. □

We briefly introduce the notion of “Zassenhaus ideals”, which will be needed in the proof of the next lemma. Let G be an arbitrary group. The Lie powers of the augmentation ideal I(kG) are the (two-sided) ideals Δ^{(i)}(kG) defined inductively by

Δ^{(1)}(kG) = I(kG), Δ^{(i+1)}(kG) = [I(kG), Δ^{(i)}(kG)]kG,

where [M, N] denotes the k-submodule generated by all elements mn − nm with m ∈ M and n ∈ N. Following [14, 24] we define the Zassenhaus ideals H_{n}(kG) of kG for n ≥ 1 by setting

H_{n}(kG) = ∑_{ip^{j}≥n} Δ^{(i)}(kG)^{p^{j}} + I(kG)^{n+1}. (3.4)

These ideals have an alternative description in terms of the Brauer–Jennings– Zassenhaus series for G, called M-series by Jennings (see [13], or [15, p. 481]): M_{1}(G) = G, and for n ≥ 2

M_{n}(G) = ⟨[G, M_{n−1}(G)], M_{⌈n/p⌉}(G)^{p}⟩ (3.5)

where ⌈n/p⌉ is the smallest integer ≥ n/p. Jennings’ result is that

M_{n}(G) = D_{n}(G), (3.6)

where the n-th dimension subgroup D_{n}(G) is defined by

D_{n}(G) = G ∩ (1 + I(kG)^{n}).

Passi and Sehgal [14] showed that

\begin{aligned}H_{n}(kG) & = I(kM_{n}(G))kG + I(kG)^{n+1} \\ & = (M_{n}(G) − 1) + I(kG)^{n+1}. \end{aligned} ; (3.7)

Now we are in a position to prove:

Lemma 3.5. Let G be a finite p-group with a proper normal subgroup A of order p. If G is not a Klein’s four-group, and J is a 1-dimensional ideal in s(G, A) such that s(G, A)/J is the group algebra of a finite group, then J = {A − 1}.

Proof. Let J be a 1-dimensional ideal in s(G, A) such that s(G, A)/J is the group algebra of a finite group. As s(G, A) modulo its radical is isomorphic to k, it follows that s(G, A)/J is isomorphic to kP for a p-group P. Let J_{0} be the kernel of the natural homomorphism kG s(G, A)/J, let π : kG kG/J_{0} be the natural map, and fix an isomorphism θ : kG/J_{0} kP. Suppose that J ≠ {A−1}. Then (A − 1)π ≠ 0. As {A − 1} annihilates the radical of s(G, A), the image (A − 1)πθ annihilates the radical of kP, and it follows that k(A−1)πθ = kP̂, where P̂ denotes the sum of the elements of P. We have A ⊆ M_{n}(G) \ M_{n+1}(G) for some integer n; by (3.6), equivalently A − 1 ∈ I(kG)^{n} \ I(kG)^{n+1}.

Suppose that I(kP)^{n+1} ≠ 0, so that P̂ ∈ I(kP)^{n+1}. Then A−1 ⊆ I(kG)^{n+1} +J_{0}. Note that J, as a 1-dimensional ideal in s(G, A), is contained in the annihilator of the radical of s(G, A), so by Lemma 3.4 and (3.3), J_{0} = I(kA)I(kG) + kσ where σ denotes the sum of a system of coset representatives of A in G. It follows that A − 1 ⊆ I(kG)^{n+1} + kσ. Consequently σ ∈ I(kG)^{n} \ I(kG)^{n+1} and

kσ + I(kG)^{n+1} = (A − 1) + I(kG)^{n+1}. (3.8)

However, we can choose a Jennings basis of I(kG) (see [13], or [15, Chapter 3, §3]) which, by [9, Theorem 7], contains the sum σ̃ of a system of coset representatives of A in G, and by (3.3), σ̃ ∈ σ + (A − 1) + I(kG)^{n+1}. We can also assume that the Jennings basis contains a − 1 for some 1 ≠ a ∈ A. Then σ̃ and a − 1 are linearly independent over I(kG)^{n+1} (see [15, Chapter 3, Lemma 3.5]), which contradicts (3.8). Hence I(kP)^{n+1} = 0, and I(kP)^{n} = kP̂.

We have shown that I(kG)^{n+1} is contained in J_{0}, and that I(kG)^{n} maps to a 1-dimensional ideal in kG/J_{0}. Together with (3.7), it follows that the Zassenhaus ideal H_{n}(kG) maps under π onto (A − 1)π, so H_{n}(kG)πθ = kP̂. Furthermore, H_{n}(kG)πθ = H_{n}(kP) by (3.4) since I(kG)πθ = I(kP). By (3.7), H_{n}(kP) = M_{n}(P) − 1. Hence kP̂ = M_{n}(P) − 1 and it follows that n = 1, I(kP)^{2} = 0, kP = 𝔽_{2}C_{2} (= group ring of a cyclic group of order 2), and |G| = 4. Thus G is cyclic (by assumption), and J = {A − 1} since s(G, A) = 𝔽_{2}C_{4}/I(𝔽_{2}C_{4})^{3} has only one ideal of dimension 1. This final contradiction proves the lemma. □

We note that the case of G being a Klein’s four-group must be excluded. We also note that if G is a Klein’s four-group, then with A = G and B a subgroup of order 2, we have s(G, A) ≅ s(G, B) as I(𝔽_{2}A)I(𝔽_{2}G) = I(𝔽_{2}G)^{2} = I(𝔽_{2}B)I(𝔽_{2}G).

We finally record:

Proposition 3.6. Let G and H be finite non-cyclic p-groups, with central elementary abelian proper subgroups A and B, respectively. Suppose that H/B is not of order two. Then an isomorphism between s(G, A) and s(H, B), if existing, maps {A − 1} onto {B − 1}.

Proof. Assume that we are given an isomorphism θ : s(G, A) s(H, B). For some set T of coset representatives of A in G, let τ_{A} be the image of ∑_{t∈T} t in s(G, A), and define similarly τ_{B} ∈ s(G, B).

Suppose that A = 1 and B ≠ 1. Then the annihilator of the radical of s(G, A) (= kG) is 1-dimensional. Consequently {B − 1} is 1-dimensional and τ_{B} does not annihilate the radical of s(H, B), since otherwise the annihilator of the radical of s(H, B) would be at least 2-dimensional. By Lemma 3.4, it follows that H is cyclic, contrary to our assumption. Thus we can assume that A, B ≠ 1.

Suppose that τ_{A} annihilates the radical of s(G, A) but τ_{B} does not annihilate the radical of s(H, B). Then by Lemma 3.4, {A−1}θ = {B̃ −1} for some subgroup B̃ of B of index p. By (3.2), it follows that kG/A ≅ s(H/B̃, B/B̃). But kG/A has dimension |G/A| whereas s(H/B̃, B/B̃) has dimension 1 + |H/B|, which is impossible since p divides the first, but not the second number. Hence we can assume, by Lemma 3.4, that both τ_{A} and τ_{B} annihilate the radicals.

By Lemma 3.4, θ maps {A − 1} + kτ_{A} onto {B − 1} + kτ_{B}. In particular, A and B have the same order. There are subgroups Ã and B̃ of index p in A and B, respectively, such that θ maps {Ã − 1} onto {B̃ − 1}. By (3.2), θ induces an isomorphism between the small group rings s(G/Ã, A/Ã) and s(H/B̃, B/B̃). The image J of {A/Ã − 1} in s(H/B̃, B/B̃) is a 1-dimensional ideal, with quotient isomorphic to the group algebra kG/A. By assumption, H/B̃ is not a Klein’s fourgroup, so J = {B/B̃ − 1} by Lemma 3.5, and it follows that θ maps {A − 1} onto {B − 1}. □

Corollary 3.7. Let G and H be finite p-groups, with central elementary abelian subgroups A and B contained in the Frattini subgroups Φ(G) and Φ(H), respectively. If G is cyclic, suppose that G and H have the same order. Then an isomorphism between s(G, A) and s(H, B), if existing, maps {A − 1} onto {B − 1}.

Proof. The radical of s(G, A) modulo its square is isomorphic to I(kG)/I(kG)^{2}, a quotient which is known to be isomorphic to G/Φ(G) (cf. [10, VI, Lemma 4.1]).

It follows that G and H are either both cyclic or non-cyclic. If H is non-cyclic, then H/B is not of order two, and the statement follows from Proposition 3.6. Otherwise G and H are cyclic of the same order, and the small group rings are uniserial (see [9, Corollary 14] or [1, p. 26]), so the statement follows also in this case. □

4. Central Frattini extensions and the small group ring

Let H be a (fixed) finite p-group, and let

[figure: 1; R; F; H; 1] (4.1)

be any presentation with F free. Additionally, we assume that

R ≤ [F, F]F^{p} (4.2)

(this holds, for example, if F is free on a minimal generating set of H). We set

L = [F, R]R^{p}

and E = F/L, V = R/L, so that

[figure: 1; R/L; F/L; H; 1; 𝔼 :; 1; V; ‖; E; ‖; H; ‖; 1] (4.3)

is a central extension of H by the elementary abelian p-group V, and by (4.2), V is contained in the Frattini subgroup Φ(E). We shall see (in Lemma 4.2) that 𝔼 is the ‘universal one’ among all extensions of H having these properties.

Definition 4.1. We denote by 𝔽(H) the collection of all central extensions

[figure: 1; A; G; H; 1]

with A an elementary abelian p-group whose image in G is contained in the Frattini subgroup of G. (Such an extension might be called a central Frattini extension.)

In this section, we give a description of the isomorphism classes of the extensions in 𝔽(H) in terms of an Out(H)-action on V, as well as a description of a coarser equivalence relation related to the small group ring in terms of an Out(kH)-action on V. (As before, k denotes the field with p elements.) We begin with the following basic lemma.

Lemma 4.2. Suppose that an exact sequence in 𝔽(H) is given:

[figure: 1; A; ι; G; φ; H; 1.] (4.4)

Then G ≅ E/obspace 𝔼_{φ}, where

[figure: 𝔼_{φ} :; 1; V; E; H; 1.; G; φ] (4.5)

More precisely, we have a commutative diagram in which all vertical maps are isomorphisms:

[figure: 1; A; ι; G; φ; H; 1]

[figure: 1; V/obspace 𝔼_{φ}; E/obspace 𝔼_{φ}; H; ‖; 1.]

Proof. By the universal property of F, we have a commutative diagram

F

[figure: 1; A; ι; G; φ; ψ; H; 1]

1

Note that G is generated by Aι and F ψ. Since Aι ≤ Φ(G), and elements of the Frattini subgroup are non-generators (see [17, 5.2.12]), it follows that ψ is surjective. Moreover, Rψ ≤ Aι implies that Lψ = [F, R]R^{p}ψ = 1, so that we have a factorization

\begin{aligned}& & & E \\ G & \overline{\begin{matrix}φ \\ \end{matrix}} & \underline{} & H & \end{aligned}

Now consider the commutative diagram

1  1

[figure: 1; A; G; φ; H; 1]

[figure: 𝔼_{φ} :; 1; V; E; H; ‖; 1]

[figure: K; ∪; = = = =; K; ∪; G; φ]

[figure: 1; 1]

Set U = obspace 𝔼_{φ}. By Proposition 2.2, U ≤ K. In particular, |G| = |E/K| ≤ |E/U|. The same argument as in the beginning of the proof shows that a lifting G E/U of φ is surjective, so |G| ≥ |E/U|. Thus U = K as desired, and the final statement follows from the above diagram. □

Next, we define two equivalence relations on 𝔽(H).

Definition 4.3. Two extensions

[figure: 1; Ã; ι̃; G̃; φ̃; H; 1,; 1; A; ι; G; φ; H; 1] (4.6)

from 𝔽(H) are isomorphic if there exists a commutative diagram

[figure: 1; Ã; ι̃; G̃; φ̃; H; 1; 1; A; ι; G; φ; H; α; 1] (4.7)

where the vertical maps are isomorphisms. Furthermore, we shall say that the two extensions (4.6) are s-equivalent if we have a commutative diagram

[figure: 0; {Ã − 1}; s(G̃, Ãι̃); φ̃; kH; 0; 0; {A − 1}; s(G, Aι); φ; kH; α; 0] (4.8)

where the vertical maps are isomorphisms.

Certainly, isomorphic extensions are s-equivalent. If we consider short exact sequences (4.6) from 𝔽(H) up to one of these equivalence relations, we can assume (and will do) that the injections ι and ι̃ are in fact inclusions.

By Corollary 3.7, two extensions (4.6) from 𝔽(H) are s-equivalent if and only if the associated small group rings s(G, A) and s(G̃, Ã) are isomorphic.

We shall define an action of Out(kH) on V = R/L = R/[F, R]R^{p}. As H is a p-group, Out(H) naturally embeds into Out(kH), a fact which was observed independently by Coleman [5] and Ward [22]. Then we will have the following straightforward result.

Theorem 4.4. The isomorphism classes of extensions in 𝔽(H) are in bijection with the Out(H)-orbits of subspaces of V, and the s-equivalence classes of extensions in 𝔽(H) are in bijection with the Out(kH)-orbits of subspaces of V. Both correspondences are given as follows: The class of the extension (4.4) in 𝔽(H) corresponds to the orbit of obspace 𝔼_{φ}, where the diagram 𝔼_{φ} is given by (4.5). Conversely, the orbit of U ≤ V corresponds to the class of

[figure: 1; V/U; E/U; H; 1.]

The action of Out(kH). We shall first define the action of Out(H) on V, which can be done more easily. Let α ∈ Aut(H). By the universal property of F, we have a commutative diagram

[figure: 1; R/L; F/L; H; 1; 1; R; F; β; H; α; 1]

From Rβ ⊆ R/L it is immediate that Lβ = [F, R]R^{p}β = 1, and we obtain a ‘lift’ α̂, i.e., a commutative diagram

[figure: 1; V; E; H; 1; 1; V; E; α̂; H; α; 1]

Note that α̂ ∈ Aut(E) since V ≤ Φ(E). Assume that for some γ ∈ Aut(E), we have a commutative diagram

[figure: 1; V; E; H; 1; 1; V; E; γ; H; ‖; 1]

This means that γ induces the identity on E/V. A simple computation using that V is central of exponent p shows that γ fixes Φ(E) = [E, E]E^{p}, so in particular V, element-wise. Thus we get a well-defined action of Aut(H) on V by letting α act via a lift α̂. Inner automorphisms lift to inner automorphisms which act trivially on V, so we really have an action of Out(H) on V.

Similarly, the action of Out(kH) on V is defined. Let α ∈ Aut(kH). By the universal property of F, we have a commutative diagram

[figure: 1; V; s(E, V)^{×}; (kH)^{×}; 1]

[figure: 1; 1 + I(kV)kE; (kE)^{×}; π; (kH)^{×}; ‖; 1]

[figure: 1; R; F; β; H; α; 1]

The k-linear extension β : kF kE maps I(kR)kF into I(kV)kE and I(kF) into I(kE). Also note that V is a central elementary abelian p-subgroup of s(E, V)^{×}, so Lβπ = 1. Hence we get a lift α̂ such that

[figure: 0; {V − 1}; s(E, V); kH; 0; 0; {V − 1}; s(E, V); α̂; kH; α; 0] (4.9)

is commutative. Let α̂^{−1} be a lift of α^{−1}. Then Eα̂α̂^{−1} maps onto H under the map s(E, V) kH, whence is a subgroup of E. Since V ≤ Φ(E), it follows that Eα̂α̂^{−1} = E, and consequently α̂ ∈ Aut(s(E, V)).

Assume that for some γ ∈ Aut(s(E, V)), we have a commutative diagram

[figure: 0; {V − 1}; s(E, V); kH; 0; 0; {V − 1}; s(E, V); γ; kH; ‖; 0]

Then γ fixes [E, E]E^{p}, so in particular V, element-wise. Using the natural isomorphism V ≅ {V − 1}, we obtain a well-defined action of Aut(kH) on V by letting α act via a lift α̂. Inner automorphisms lift to inner automorphisms which act trivially on {V − 1}, so we have indeed an action of Out(kH) on V.

It is immediate that this action, when restricted to Out(H), coincides with the action of Out(H) defined before.

Proof of Theorem 4.4. We first treat the parametrization of the isomorphism classes of extensions. Given two isomorphic extensions in 𝔽(H), as shown in (4.7), and any subspace U of V, we have a commutative diagram

[figure: 𝔼φ̃ :; V; ⊂; E; H]

[figure: 𝔼_{φ} :; V; ⊂; E; α̂; H; α]

Uα̂^{∪} = = = = = = = = Uα̂^{∪}  G̃^{φ̃ }

[figure: U; ∪; = = = = = = = = = = = = =; U; ∪; G; φ; ≅]

showing that φ factors over E/U if and only if φ̃ factors over E/Uα̂. Thus α̂ maps obspace 𝔼_{φ} to obspace 𝔼φ̃ by Proposition 2.2, and both extensions define the same Out(H)-orbit of subspaces of V.

Conversely, let U be a subspace of V, and let α ∈ Aut(H), with lift α̂ ∈ Aut(E). Then we have a commutative diagram

[figure: 1; V/Uα̂; E/Uα̂; H; 1; 1; V/U; E/U; H; α; 1]

where the vertical map in the middle is induced by α̂. This shows that U and Uα̂ define isomorphic extensions. Thus we have well-defined maps, and it follows from Lemma 4.2 that they are mutually inverse.

Next, assume that we are given two extensions in 𝔽(H), as shown in (4.6), which are s-equivalent, as shown in (4.8). Then for any subspace U of V, we have a commutative diagram

[figure: {V −1}; ⊂; s(E, V); kH]

[figure: {V −1}; ⊂; s(E, V); α̂; kH; α]

\{Uα̂−1\}^{∪}  = = = = {= \atop =} \{Uα̂−1\}^{∪}  s(G̃, Ã)^{φ̃ }

[figure: {U −1}; ∪; = = = = = = = =; {U −1}; ∪; s(G, A); φ; ≅]

Hence φ factors over s(E, V)/{U −1} if and only if φ̃ factors over s(E, V)/{Uα̂−1}.

Note that a commutative diagram

[figure: 1; V/U; E/U; H; 1; G; φ]

affords by (3.2) a commutative diagram

[figure: 0; \frac{\{V −1\}}{\{U −1\}}; \frac{s(E, V)}{\{U −1\}}; kH; 0]

[figure: 0; {V/U −1}; s(E/U, V/U); kH; ‖; 0]

[figure: kG; φ]

Since I(kA) is mapped into {V/U − 1} under the diagonal map, I(kA)I(kG) is contained in the kernel of the diagonal map, and there is an induced commutative diagram

[figure: 0; \frac{\{V −1\}}{\{U −1\}}; \frac{s(E, V)}{\{U −1\}}; kH; 0; s(G, A); φ]

The same applies to the homomorphism φ̃ : G̃ H.

Thus α̂ maps {obspace 𝔼_{φ} − 1} to {obspace 𝔼φ̃ − 1} by Corollary 3.2 (where 𝔼_{φ} is the diagram (4.5), and 𝔼φ̃ is defined analogously), and both extensions define the same Out(kH)-orbit of subspaces of V.

Conversely, let U be a subspace of V, and let α ∈ Aut(kH), with lift α̂ ∈ Aut(s(E, V)). By (3.2), we have a commutative diagram

[figure: 0; {V/Uα̂−1}; s(E/Uα̂, V/Uα̂); kH; 0]

[figure: 0; \frac{\{V −1\}}{\{Uα̂−1\}}; \frac{s(E, V)}{\{Uα̂−1\}}; kH; ‖; 0]

[figure: 0; \frac{\{V −1\}}{\{U −1\}}; \frac{s(E, V)}{\{U −1\}}; kH; α; 0]

[figure: 0; {V/U −1}; s(E/U, V/U); kH; ‖; 0]

where the vertical map in the middle is induced by α̂, and all vertical maps are isomorphisms. Hence the canonical sequences

[figure: 1; V/Uα̂; E/Uα̂; H; 1,; 1; V/U; E/U; H; 1]

are s-equivalent. Again we have well-defined maps, which are mutually inverse by Lemma 4.2, and the proof of Theorem 4.4 is complete. □

We conclude this section with a couple of remarks.

Remark 4.5. We have shown that any automorphism of kH can be lifted to an automorphism of s(E, V). Conversely, any automorphism of s(E, V) induces an automorphism of kH by Corollary 3.7. An automorphism of s(E, V) which induces the identity of kH is really an automorphism of E which induces the identity on E/V. For the moment, let X denote the group of all these automorphisms. Any σ in X fixes Φ(E) element-wise (this we noticed already when proving that the Aut(H)-action on V is well-defined), and gives rise to a homomorphism E V, x ↦ x^{−1}(xσ). In this way, we obtain a homomorphism X Hom(E/Φ(E), V), and since conversely any map from Hom(E/Φ(E), V) gives rise to an automorphism in X, this is an isomorphism (of abelian groups). Thus we have an exact sequence

[figure: 1; Hom(H/Φ(H), V); Aut(s(E, V)); Aut(kH); 1.]

Remark 4.6. Let U ≤ V. The extensions from 𝔽(H) which are s-equivalent to

[figure: 1; V/U; E/U; H; 1]

can also be described as follows. We may start from the exact sequence for the small group ring s(E/U, V/U), with associated exact sequence of unit groups

[figure: 1; V/U; s(E/U, V/U)^{×}; κ; (kH)^{×}; 1.]

Then the extensions

[figure: 1; V/U; κ^{−1}(Hα); κ; Hα; 1,]

α ∈ Aut(kH), yield all isomorphism classes of extensions from 𝔽(H) which are s-equivalent to the above extension.

We shall see in the next section that the middle terms κ^{−1}(Hα) need not be isomorphic to E/U, which shows that automorphisms of kH in general cannot be lifted to automorphisms of s(E/U, V/U).

To verify the claim, let α ∈ Aut(kH), with lift α̂ ∈ Aut(s(E, V)). Then we have an isomorphism

[figure: 1; V; Eα̂; Hα; 1; 1; V; E; α̂; H; α; 1]

which together with (3.2) shows that we have isomorphisms

[figure: 1; V/U; κ^{−1}(Hα); κ; Hα; 1]

[figure: 1; V/U; ‖; Eα̂/U; Hα; ‖; 1]

[figure: 1; V/Uα̂^{−1}; E/Uα̂^{−1}; α̂; H; α; 1]

For computational purposes, however, the description of an s-equivalence class given by Theorem 4.4 is more convenient.

Remark 4.7. We found Theorem 4.4 while thinking about how the attempt from [2, 3] might be substantiated, as we will briefly set out. Assume that we are given an exact sequence in 𝔽(H):

[figure: 1; A; ⊂; G; φ; H; 1.]

Then we have a diagram

[figure: 0; {V − 1}; s(E, V); kH; 0; kG; φ]

By Corollary 3.2 and Lemma 4.2, there is a unique minimal ideal I = {U − 1} of s(E, V) contained in {V −1} such that φ factors over s(E, V)/I, and φ : G H does not only factor over E/U (the pre-image of H), but in fact G ≅ E/U.

Assume that we are given another exact sequence in 𝔽(H):

[figure: 1; Ã; ⊂; G̃; φ̃; H; 1,]

and that G̃ is a group basis of kG. Then, of course, similar remarks hold for the homomorphism φ̃ : kG̃ = kG kH. Suppose that Gφ̃ is a group basis of kH, isomorphic to H. Then there is an automorphism α of kH making the following diagram commutative:

[figure: kH; α; kH; kG; φ; = = =; kG̃; φ̃] (4.10)

We have seen that α lifts to an automorphism α̂ of s(E, V), so that we have a commutative diagram

[figure: {V −1}; ⊂; s(E, V); kH]

[figure: {V −1}; ⊂; s(E, V); α̂; kH; α]

Iα̂^{∪} = = = = = = = = = = Iα̂^{∪}  kG̃^{φ̃ }

[figure: I; ∪; = = = = = = = = = = = = =; I; ∪; kG; φ; = = = = = =]

It follows that φ̃ factors over s(E, V)/Iα̂ with Iα̂ = {U − 1}α̂ = {Uα̂ − 1}, and that G̃ ≅ E/Uα̂. Since we wish to conclude that G ≅ G̃, we would like to have the subspace I to be fixed under α̂. One might hope that the action of α̂ can be compensated by a group automorphism σ of H (which lifts to an automorphism σ̂ of E, as we have seen): Assuming that Iα̂σ̂^{−1} = I, we have a commutative diagram

[figure: {V −1}; ⊂; s(E, V); kH]

[figure: {V −1}; ⊂; s(E, V); _{α̂_{σ̂^{−1}}}; kH; ασ^{−1}]

I^{∪} = = = = = = = = = = = = I^{∪}  kG̃^{φ̃σ^{−1} }

[figure: I; ∪; = = = = = = = = = = = = =; I; ∪; kG; φ; = = = = = =]

and can conclude that φ̃σ^{−1} factors over s(E, V)/I, so that G̃ ≅ E/U ≅ G (note that E is still the pre-image of G̃φ̃σ^{−1}). But it appears that all this is asked for too much.

5. First examples

Throughout this section we stick to the notation introduced in the last section. Moreover, the reader should be aware of the notion of unipotent automorphisms (cf. Definition 6.1).

Still, p denotes a (fixed) prime number and k the field with p elements.

Example 5.1 (Cyclic groups). It is certainly instructive to start by considering cyclic groups, so let H = C_{p^{n}} = ⟨x_{n}⟩ be a cyclic group of order p^{n} for some n ∈ ℕ. The obvious presentation with F = ⟨x⟩, R = ⟨x^{p^{n}}⟩ and surjection F H defined by x ↦ x_{n} gives rise to the following ‘universal’ central Frattini extension where L = R^{p} = ⟨x^{p^{n+1}}⟩, E = F/L = ⟨x_{n+1}⟩, V = R/L = ⟨x_{n+1}^{p^{n}}⟩ and π is defined by x_{n+1} ↦ x_{n}:

[figure: 𝔼 :; 1; V; ⊂; E; π; H; 1; 1; C_{p}; ≅; ⊂; C_{p^{n+1}}; ‖; C_{p^{n}}; ‖; 1]

The substitution X ↦ x_{n} − 1 induces a k-algebra isomorphism between kH and the truncated polynomial ring k[X̄] = k[X]/(X^{p^{n}}), which identifies the radical I(kH) with the ideal generated by X̄ = X̄_{n} = X + (X^{p^{n}}). Therewith, the ideal I(kV) = ⟨x_{n+1}^{p^{n}} − 1⟩ = ⟨(x_{n+1} − 1)^{p^{n}}⟩ within kE is identified with (X̄_{n+1}^{p^{n}}) and the small group ring is nothing but a truncated polynomial ring again,

s(E, V) = \frac{kE}{I(kV)I(kE)} ≅ \frac{k[X]}{(X^{p^{n}+1})}.

Note that {V − 1} corresponds to the ideal (X̄^{p^{n}}).

Any automorphism of the group algebra k[X̄_{n}] is induced from a substitution X̄_{n} ↦ λX̄_{n} + 𝔯 with λ ∈ k^{×} and 𝔯 ∈ (X̄_{n}^{2}). The unipotent automorphisms are precisely the maps arising from substitutions with λ = 1, whereas the maps arising from substitutions with 𝔯 = 0 constitute a subgroup S isomorphic to k^{×} which complements the group UAut(k[X̄_{n}]) of unipotent automorphisms. Thus, Aut(k[X̄_{n}]) ≅ UAut(k[X̄_{n}]) ⋊ k^{×}. Any automorphism of H can be described by x_{n} ↦ x_{n}^{β} for some β ∈ ℕ relatively prime to p, and gives rise to the substitution

X̄_{n} = (X̄_{n} + 1) − 1 (X̄_{n} + 1)^{β} − 1 = ∑_{i=1}^{β} \binom{β}{i}X̄_{n}^{i} ∈ βX̄_{n} + (X̄_{n}^{2})

which can always be compensated by a map from S to obtain a unipotent automorphism. Thus, we see directly that Aut(kH) = Aut(H) · UAut(kH).

Unipotent automorphisms of s(E, V), arising from substitutions X ↦ X +𝔯 with 𝔯 ∈ (X^{2}), induce the identity on {V − 1}:

X^{p^{n}} (X + 𝔯)^{p^{n}} = X^{p^{n}} + 𝔯^{p^{n}} ∈ X^{p^{n}} + (X^{p^{n}+1}).

On the other hand, decomposing any group automorphism along S · UAut(k[X̄]) shows that Aut(H) acts just as k^{×} on {V − 1}:

X^{p^{n}} (λX)^{p^{n}} = λ^{p^{n}}X^{p^{n}} = λ · X^{p^{n}} for λ ∈ k^{×}.

Note that Out(kH) = Aut(kH) and Out(H) = Aut(H) since H is abelian. Thus the actions of Out(kH) and Out(H) coincide, and correspond to multiplication with field elements from k^{×} on the 1-dimensional space {V − 1}. This observation (with n = 1) will be useful in the following example when considering diagonal matrices.

Example 5.2 (Elementary abelian groups). Let H ≅ C_{p}^{r} be an elementary abelian p-group of rank r ≥ 2 (the case r = 1 is included in Example 5.1). We choose the obvious presentation for H with free group F = ⟨x_{1}, …, x_{r}⟩ and relations

R = ⟨x_{i}^{p}, [x_{i}, x_{j}] | 1 ≤ i, j ≤ r, i < j⟩.

The ‘universal’ central Frattini extension with quotient group H is given by E = F/L where L = [F, R]R^{p}. We set x̄_{i} := x_{i}R ∈ H and x̂_{i} := x_{i}L ∈ E. Observe that the rank of V is precisely r + \binom{r}{2}, with a minimal set of generators given by {x̂_{i}^{p}, [x̂_{i}, x̂_{j}] | 1 ≤ i, j ≤ r, i < j}.

If we want to parametrize the isomorphism and s-equivalence classes of extensions in 𝔽(H) then, by Theorem 4.4, we have to understand the action of Out(kH) on V. Note that R = [F, F]F^{p} = M_{2}(F), so H = F^{[1]} in the notation from Section 6, and we have a factorization Aut(kH) = Aut(H)·UAut(kH), by Lemma 6.4.

First we are going to show that the group of unipotent automorphisms UAut(kH) acts trivially on V. Therefore, the actions of Out(kH) and Out(H) on V coincide, implying that isomorphism classes in 𝔽(H) are in fact s-equivalence classes. Note that only for the prime p = 2 we have L = M_{3}(F) and, thus, M_{3}(E) = 1, so we cannot deduce from Corollary 6.5 the trivial action of unipotent automorphisms.

The key equation implying the trivial action of unipotent automorphisms is the following:

\begin{matrix}x̂_{i}(x̂_{j} − 1) − (x̂_{j} − 1)x̂_{i} \\ = (x̂_{i} − 1)(x̂_{j} − 1) − (x̂_{j} − 1)(x̂_{i} − 1) \\ = x̂_{i}x̂_{j} − x̂_{j}x̂_{i} \\ = −x̂_{i}x̂_{j}([x̂_{j}, x̂_{i}] − 1) ∈ I(kV)kE. \end{matrix} ; (5.1)

Therefore, we have modulo I(kV)I(kE) that

\begin{matrix}x̂_{i}(x̂_{l} − 1) ⋯ (x̂_{m} − 1) ≡ (x̂_{l} − 1)x̂_{i} ⋯ (x̂_{m} − 1) \\ ≡ … ≡ (x̂_{l} − 1) ⋯ (x̂_{m} − 1)x̂_{i}, \end{matrix} ; (5.2)

showing that the image of x̂_{i} in the small group ring s(E, V) commutes with all monomials of degree ≥ 2 in the images of the x̂_{j} − 1 (which generate the square of the radical). This immediately implies that modulo I(kV)I(kE), we have:

(5.3)  ((x̂_{i} − 1)(x̂_{j} − 1))^{p} ≡ (x̂_{i} − 1)^{p}(x̂_{j} − 1)^{p} = (x̂_{i}^{p} − 1)(x̂_{j} − 1)^{p} ≡ 0.

By slight abuse of notation, we denote the image of x̂_{i} in s(E, V) again by x̂_{i}. Let α ∈ UAut(kH), and consider a lift α̂ ∈ Aut(s(E, V)), which can be described by

x̂_{i}α̂ = x̂_{i} + 𝔯_{i},

where 𝔯_{i} ∈ I(kE)^{2}/I(kV)I(kE). Combining Equations (5.2) and (5.3) we obtain:

x̂_{i}^{p}α̂ = (x̂_{i} + 𝔯_{i})^{p} = x̂_{i}^{p} + 𝔯_{i}^{p} = x̂_{i}^{p}.

Using again the commutation relation (5.2), we can check directly that α̂ fixes the element (x̂_{i} − 1)(x̂_{j} − 1) − (x̂_{j} − 1)(x̂_{i} − 1). Since by (5.1), the additive commutator (x̂_{i} − 1)(x̂_{j} − 1) − (x̂_{j} − 1)(x̂_{i} − 1) and [x̂_{i}, x̂_{j}] − 1 represent the same element in s(E, V), we see that α̂ fixes the elements of a minimal generating set of V. Thus α acts trivially, as we wished to show.

Note that Out(H) = Aut(H) ≅ GL_{r}(k): A matrix A = (a_{ij})_{i,j=1}^{r} ∈ GL_{r}(k) corresponds to the automorphism α_{A} of H given by x̄_{i} ↦ ∏_{j=1}^{r} x̄_{j}^{a_{ij}}. The obvious lift x_{i} ↦ ∏_{j=1}^{r} x_{j}^{a_{ij}} of α_{A} to F induces a lift α̂_{A} ∈ Aut(E) leaving V invariant, thus giving rise to the GL_{r}(k)-action under consideration. Next, we determine the action of Aut(H) on V, i.e., the structure of V as GL_{r}(k)-module. Let W = k^{r} be the natural GL_{r}(k)-module with canonical basis {w_{i} | 1 ≤ i ≤ r}.

First, we note that the subgroup T of V generated by the commutators [x̂_{i}, x̂_{j}] for i < j is a GL_{r}(k)-submodule. Since the group GL_{r}(k) is generated by elementary and diagonal matrices, it is enough to consider A = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(k). Using basic commutator relations (see [11, Chapter III, (1.2)]), and the definition of L, we obtain:

[x̂_{i}, x̂_{j}]α̂_{A} = [x̂_{i}^{a}x̂_{j}^{b}, x̂_{i}^{c}x̂_{j}^{d}] = [x̂_{i}^{a}x̂_{j}^{b}, x̂_{j}^{d}] · [x̂_{i}^{a}x̂_{j}^{b}, x̂_{i}^{c}]
= [x̂_{i}^{a}, x̂_{j}^{d}] · [x̂_{j}^{b}, x̂_{i}^{c}] = [x̂_{i}, x̂_{j}]^{ad} · [x̂_{j}, x̂_{i}]^{bc}
= [x̂_{i}, x̂_{j}]^{ad−bc} = [x̂_{i}, x̂_{j}]^{det(A)}.

This shows, in addition, that w_{i}∧w_{j} ↦ [x̂_{i}, x̂_{j}] (i < j) defines a GL_{r}(k)-isomorphism between ⋀^{2} W and T. Observing that E is a nilpotent group of class 2 and making use of [11, Chapter III, (1.3)], we obtain the action of the matrix A on the missing generators x̂_{i}^{p} (1 ≤ i ≤ r) of V:

x̂_{i}^{p}α̂_{A} = (x̂_{i}^{a}x̂_{j}^{b})^{p}

= (x̂_{i}^{a})^{p}(x̂_{j}^{b})^{p}[x̂_{j}^{b}, x̂_{i}^{a}]^{\binom{p}{2}}

= \begin{cases}(x̂_{i}^{p})^{a}(x̂_{j}^{p})^{b} & for p odd, \\ (x̂_{i}^{p})^{a}(x̂_{j}^{p})^{b} · [x̂_{i}, x̂_{j}]^{ab} & for p = 2. \end{cases}

Thus, in any case, V/T is isomorphic to W as GL_{r}(k)-module, an isomorphism being given by w_{i} ↦ x̂_{i}^{p}T. So we have a short exact sequence of GL_{r}(k)-modules

[figure: 𝔼_{r} :; 0; ⋀^{2} W; V; W; 0,]

the sequence being split for odd primes p.

Let p = 2, and write more clearly V_{r} = V. The sequence 𝔼_{2} turns out to be split by sort of an accident: generators of GL_{2}(𝔽_{2}) (isomorphic to the symmetric group on 3 letters) act on V_{2} via the matrices ( \begin{matrix}1 & 1 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) and ( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ), which can be

conjugated into block diagonal form by ( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ). But 𝔼_{r} for r ≥ 3 is a non-split extension of GL_{r}(k)-modules: For r = 3, this can be shown by direct inspection, and for r > 3, this can be deduced inductively, just noting that the restriction of the GL_{r}(k)-module V_{r} to GL_{r−1}(k) = Stab_{GL_{r}(k)}(w_{r}) splits according to (in suggestive notation)

V_{r}|_{GL_{r−1}(k)} = V_{r−1} ⊕ (V_{r−1} ∧ w_{r}) ⊕ kw_{r}.

Therefore, a splitting of 𝔼_{r} would induce a splitting of 𝔼_{r−1}, which is impossible by the induction hypothesis.

Summarizing, we have reduced the parametrization of s-equivalence (= isomorphism) classes of central Frattini extensions with elementary abelian factor p-group of rank r and middle group of order p^{2r+\binom{r}{2}−d} to the combinatorial determination of GL_{r}(k)-orbits of d-dimensional subspaces of the (r + \binom{r}{2})-dimensional GL_{r}(k)- module V_{r} from the sequence 𝔼_{r}.

Example 5.3 (Non-isomorphic groups with isomorphic small group rings). In the last example we consider a case where the action of Out(kH) on V is strictly larger than the action of Out(H). Let H = D_{8} be the dihedral group of order 8 as given by the Coxeter presentation, i.e.,

H ≅ F/R = ⟨a, b | a^{2}, b^{2}, (ab)^{4}⟩.

Thus, ā = aR and b̄ = bR can be interpreted as (simple) reflections in Euclidean space ℝ^{2} generating the group of symmetries of a square.

We will omit most of the calculations, which anyway can easily be reproduced — if not by hand, then at least with the aid of a computing system like GAP (cf. [7]). For convenience, we shall use the numbering of isomorphism classes of groups of small order as established in the Small Groups library embedded in GAP.

Let L = [F, R]R^{2} and set E = F/L and V = R/L. The three relators from R give rise to a minimal generating set for V, i.e., setting â = aL and b̂ = bL, we can identify the canonical basis of 𝔽_{2}^{3} with the (ordered) set {â^{2}, b̂^{2}, (âb̂)^{4}} ⊂ V ≅ C_{2}^{3}. We shall make explicit use of this identification when representing group actions on V by matrices in GL_{3}(𝔽_{2}).

We want to parametrize the isomorphism and s-equivalence classes of extensions in 𝔽(H). By Theorem 4.4, this amounts to computing the homomorphism Out(𝔽_{2}H) GL(V) and its restriction to Out(H) (as outlined in the proof of Theorem 4.4) and studying the induced action on the set of subspaces of V. Note that M_{3}(H) = 1, i.e., H has M-length ≤ 2. Finite p-groups G with M-length ≤ 2 have a normal complement in the normalized unit group of kG (see [14, Corollary 7]), a fact that readily implies that group ring automorphisms of kG can be written as the product of a group automorphism and a unipotent automorphism. Thus Aut(𝔽_{2}H) = Aut(H) · UAut(𝔽_{2}H), and we compute both factors separately.

The symmetry of the Coxeter presentation shows that switching the two generators ā and b̄ induces an automorphism σ of H. In fact, this automorphism provides a generator of Out(H), which is a cyclic group of order two. To describe its action on V, we lift σ to the automorphism of F which switches the free generators a and b. Then, σ induces a lifting automorphism σ̂ of E which acts on V by â^{2}σ̂ = b̂^{2}, b̂^{2}σ̂ = â^{2} and

(âb̂)^{4}σ̂ = (b̂â)^{4} = â^{3} · â · (b̂â)^{4} = â^{3}(âb̂)^{4}â
= â^{3} · â · (âb̂)^{4} · [(âb̂)^{4}, â] = â^{4} · (âb̂)^{4}
= (âb̂)^{4}.

Thus, the action of a generator of Out(H) on V is determined by the permutation

matrix X = ( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ).

We set Ā = ā − 1 and B̄ = b̄ − 1. These elements generate the radical I(𝔽_{2}H) of the group ring 𝔽_{2}H. The elements of the group UAut(𝔽_{2}H) can be compactly described by

ā ↦ ā + s_{a} + t_{a}, b̄ ↦ b̄ + s_{b} + t_{b},

where s_{a}, s_{b} ∈ {0, ĀB̄ + B̄Ā}, and t_{a} and t_{b} are arbitrary elements of I(𝔽_{2}G)^{3}. This gives a total of 2^{8} unipotent automorphisms. The automorphisms τ and υ determined by letting

τ : s_{a} = ĀB̄ + B̄Ā, s_{b} = t_{a} = t_{b} = 0,
υ : s_{b} = ĀB̄ + B̄Ā, s_{a} = t_{a} = t_{b} = 0,

act on V — via liftings to automorphisms of Aut(s(E, V)) — by multiplication with the matrices Y = ( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) and Z = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ), respectively. The action of ⟨Y, Z⟩ already gives the full action of UAut(𝔽_{2}H) on V. Thus, the action of Out(𝔽_{2}H) on V is determined by the matrix group

⟨X, Y, Z⟩ = ⟨Y, Z⟩ ⋊ ⟨X⟩ ≅ (C_{2} × C_{2}) ⋊ C_{2} ≅ D_{8},

with Out(H) corresponding to the subgroup ⟨X⟩.

The 2-dimensional subspaces of 𝔽_{2}^{3} can be parametrized by all 2 × 3 matrices over 𝔽_{2} in reduced row-echelon form (cf. [12, p. 14]). There are seven of these:

M_{1} = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ) , M_{2a} = ( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ) , M_{2b} = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \end{matrix} ) , M_{3} = ( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 1 \end{matrix} ) ,

M_{4a} = ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ) , M_{5} = ( \begin{matrix}1 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , M_{4b} = ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) .

To each of these matrices M_{i} one associates an elementary abelian subgroup U_{i} of V of rank 2: For example, U_{2b} = ⟨â^{2}, b̂^{2} · (âb̂)^{4}⟩. The action of Out(H) on the 2-dimensional subspaces can be computed by calculating the reduced row-echelon form of M_{i} · X. It turns out that there are five orbits,

O_{1} = {M_{1}}, O_{2} = {M_{2a}, M_{2b}}, O_{3} = {M_{3}},
O_{4} = {M_{4a}, M_{4b}} and O_{5} = {M_{5}},

which correspond to the isomorphism classes of central Frattini extensions in 𝔽(H) with middle group of order 16, with representatives given by

[figure: 𝔼_{i} :; 1; V/U_{i}; E/U_{i}; H; 1.]

Set E_{i} = E/U_{i}. The groups E_{1}, …, E_{5} are pairwise non-isomorphic, with E_{1} of dihedral, E_{2} of semi-dihedral and E_{3} of quaternion type. But the sequences 𝔼_{1}, 𝔼_{2} and 𝔼_{3} are s-equivalent, and therefore their small group rings s(E/U_{i}, V/U_{i}) are isomorphic, as one easily sees from the action of Y: we have M_{1} ^{Y} M_{2a} and M_{2b} ^{Y} M_{3}. (It should be remarked that the series of dihedral, semi-dihedral and generalized quaternion groups are nevertheless determined by their modular group rings (cf. [4, §2]).) It turns out that the Out(H)-orbits O_{4} and O_{5} are already Out(𝔽_{2}H)-orbits.

The same kind of reasoning applied to vectors from 𝔽_{2}^{3} \ {0} yields the isomorphism and s-equivalence classes in 𝔽(H) with middle group of order 32. There are again 5 isomorphism classes of such extensions (due to the fact that X = X^{−t}), with representing vectors

v_{9} = (^{1} ^{0} ^{0} ) , v_{10} = (^{1} ^{0} ^{1} ) , v_{2} = (^{0} ^{0} ^{1} ) , v_{14} = (^{1} ^{1} ^{0} ) , v_{13} = (^{1} ^{1} ^{1} ) .

Here, the index i ∈ {2, 9, 10, 13, 14} corresponds to the number of the isomorphism class of the middle group G_{i} provided by the Small Groups library in GAP (cf. [7]). For example, we have an extension

[figure: 𝔾_{9} :; 1; V/⟨â^{2}⟩; E/⟨â^{2}⟩; H; 1]

with middle group G_{9} = E/⟨â^{2}⟩. Again, all middle groups of the sequences 𝔾_{i} are pairwise non-isomorphic. None of these is of dihedral, semi-dihedral or quaternion type. This time the classes of 𝔾_{9} and 𝔾_{10} as well as the classes of 𝔾_{13} and 𝔾_{14} are merged into one s-equivalence class, giving rise to two pairs of non-isomorphic central Frattini extensions with isomorphic corresponding small group ring sequences.

6. Röhl’s observation revisited

In this section, we discuss an observation of Röhl [19,20], thus relating it to the small group ring, and give, in our setting, a detailed exposition of the main result in [20]. We keep the notation introduced in Section 4.

Let G denote an arbitrary group.

Definition 6.1. A unipotent automorphism of kG is an augmentation-preserving automorphism of kG which induces the identity on I(kG)/I(kG)^{2}. The unipotent automorphisms of kG form a normal subgroup UAut(kG) of Aut(kG).

The notion of unipotent automorphism is also meaningful for quotients of kG. Note that a unipotent automorphism of kG induces the identity on all quotients I(kG)^{l}/I(kG)^{l+1} of successive powers of the augmentation ideal.

It is a well-known result in low degree cohomology that I(ℤG)/I(ℤG)^{2} ≅ G/[G, G] (cf. [10, VI, Lemma 4.1]; the isomorphism maps the coset of g − 1 to the coset of g, for any g ∈ G). Tensoring with k = 𝔽_{p} over ℤ yields

I(kG)/I(kG)^{2} ≅ G/[G, G]G^{p}. (6.1)

Recall that we are given a finite p-group H together with a presentation (4.1) and a ‘universal’ extension (4.3). We assume that F is free on a minimal generating set of H, so that E/[E, E]E^{p} ≅ F/[F, F]F^{p} ≅ H/[H, H]H^{p}. Combined with (6.1), this yields

I(kE)/I(kE)^{2} ≅ I(kF)/I(kF)^{2} ≅ I(kH)/I(kH)^{2}. (6.2)

Let I denote the image of I(kE) in the small group ring s(E, V) associated with the sequence (4.3). We obtain immediately:

Lemma 6.2. Let α ∈ UAut(kH), and let α̂ ∈ Aut(s(E, V)) be a lift as shown in (4.9). Then α̂ induces the identity on all sections I^{l}/I^{l+1} (i.e., α̂ is a unipotent automorphism).

Proof. We have Iα̂ = I since α is augmentation-preserving, and obtain from (4.9) an induced commutative diagram

[figure: I/I^{2}; I(kH)/I(kH)^{2}; I/I^{2}; α̂; I(kH)/I(kH)^{2}; ‖]

The horizontal maps are the isomorphism arising from (6.2) since I(kV)I(kE) ⊆ I(kE)^{2} implies that I/I^{2} ≅ I(kE)/I(kE)^{2}. Hence α̂ is a unipotent automorphism.

□

Recall the definition of the Brauer–Jennings–Zassenhaus series for the group G (and the fixed prime p) from (3.5) and (3.6). We shall write G^{[n]} = G/M_{n+1}(G) for n ≥ 1.

We record two simple lemmas.

Lemma 6.3. Suppose that H = F^{[n]} for some n ≥ 1. Then V = M_{n+1}(E), so {V − 1} is contained in I^{n+1} and {V − 1} ∩ I^{n+2} = {M_{n+2}(E) − 1}.

Proof. By assumption, R = M_{n+1}(F), so M_{n+1}(E) = M_{n+1}(F)L/L = R/L = V and the first statement follows. Let U ≤ E be such that {V − 1} ∩ I^{n+2} = {U − 1}. By (3.6), we have U = E ∩ (1 + I(kE)^{n+2} + I(kV)I(kE)) = E ∩ (1 + I(kE)^{n+2}) = M_{n+2}(E), whence the second statement. □

Next, we have (cf. [20, Theorem 3.2.1]):

Lemma 6.4. Suppose that H = F^{[n]} for some n ≥ 1. Then for every m > n,

Aut(kH/I(kH)^{m}) = Aut(H) · UAut(kH/I(kH)^{m}).

Proof. First, note that the statement makes sense since by (3.6), H embeds into kH/I(kH)^{m}, and therefore Aut(H) embeds into Aut(kH/I(kH)^{m}).

Let α ∈ Aut(kH/I(kH)^{m}). Taking augmentation is a homomorphism, so α is augmentation-preserving since H is a p-group. Hence α induces an automorphism of the abelian group I(kH)/I(kH)^{2}, which by (6.1) is canonically isomorphic to H/M_{2}(H). Thus, it suffices to show that any automorphism β of H/M_{2}(H) can be lifted to an automorphism of H. By the universal property of F, we obtain a homomorphism φ : F H making the following diagram commute:

[figure: F; φ; H; H/M_{2}(H); β; H/M_{2}(H)]

Then H = ⟨F φ, M_{2}(H)⟩ = ⟨F φ, Φ(H)⟩ = F φ (see [17, 5.2.12]). Since M_{n+1}(F) is

the kernel of any surjection F  H = F^{[n]}, we are done.  □

Corollary 6.5. Suppose that H = F^{[n]} for some n ≥ 1. Let α ∈ Aut(kH), and let α̂ ∈ Aut(s(E, V)) be a lift as shown in (4.9). Then α̂ fixes {M_{n+2}(E) − 1}, and acts on {V − 1}/{M_{n+2}(E) − 1} just like an automorphism of E.

Proof. By Lemma 6.4, Aut(kH) = Aut(H) · UAut(kH). From the description of {M_{n+2}(E) − 1} given in Lemma 6.3 it follows that α̂ fixes {M_{n+2}(E) − 1}.

Recall from the definition of the action of Out(kH) on V in Section 4 that if α ∈ Aut(H), then α actually can be lifted to an automorphism of E which acts on V just like the given lift α̂.

Thus we can assume that α is a unipotent automorphism. Then α̂ is a unipotent automorphism by Lemma 6.2, whence acts trivially on {V − 1}/{M_{n+2}(E) − 1} by Lemma 6.3. The corollary is proved. □

From now on, we let H = F^{[n]} for some n ≥ 1. Writing F_{i} = M_{i}(F), we then have R = F_{n+1}, so L = [F, R]R^{p} ≤ F_{n+2} by (3.5) and M_{n+2}(E) = F_{n+2}/L. Setting Ē = E/M_{n+2}(E), we have a central extension in 𝔽(H):

[figure: 1; F_{n+1}/F_{n+2}; F^{[n+1]}; F^{[n]}; 1; 𝔼̄ :; 1; V̄; ‖; Ē; ‖; H; ‖; 1]

For the moment, let us denote by 𝔽^{[n]} the collection of all exact sequences (4.4) in 𝔽(H) such that M_{n+2}(G) = 1.

If (4.4) is a sequence in 𝔽^{[n]}, with associated diagram 𝔼_{φ} as shown in (4.5), then M_{n+2}(E) ≤ obspace 𝔼_{φ} by Lemma 4.2, and we may replace in the formulation of Lemma 4.2 the diagram 𝔼_{φ} by the diagram 𝔼̄_{φ}. Moreover, the action of Out(kH) on V gives rise to an action of Out(kH) on V̄ which is really an Out(H)-action, as we have seen in Corollary 6.5. Following the proof of Theorem 4.4, we obtain mutatis mutandis:

Proposition 6.6. With the notation as above, the following classes are in natural bijection:

• the isomorphism classes of extensions in 𝔽^{[n]};

• the Out(H)-orbits of subspaces of V̄;

• the s-equivalence classes of extensions in 𝔽^{[n]}.

Assume that we are given two exact sequences (4.6) in 𝔽^{[n]}. These sequences are isomorphic if and only if the middle terms G and G̃ are isomorphic since A = M_{n+1}(G) and Ã = M_{n+1}(G̃). Recall from Corollary 3.7 that s-equivalence means isomorphism of small group rings. Thus we have shown that if the small group rings s(G, M_{n+1}(G)) and s(G̃, M_{n+1}(G̃)) are isomorphic, then the groups G and G̃ are isomorphic. However, as yet we do not know whether the small group rings are isomorphic provided that the group algebras kG and kG̃ are isomorphic.

If kG = kG̃, we also do not know whether we have a commutative diagram (4.10) at our disposal, so we cannot compare “obstructions” using the small group ring, but in the present case one can use the following simple observation.

Lemma 6.7. Suppose that G and G̃ are two finite groups such that G^{[n]} ≅ G̃^{[n]} for some n ≥ 1, and that there is an isomorphism θ : kG/I(kG)^{m} kG̃/I(kG̃)^{m} for some m > n+1 which maps I(kG)/I(kG)^{m} onto I(kG̃)/I(kG̃)^{m}. Set Q = G^{[n]}, and identify Q with its isomorphic image in kQ/I(kQ)^{n+1}. Then there is a commutative diagram

[figure: kQ/I(kQ)^{n+1}; α; kQ/I(kQ)^{n+1}; kG/I(kG)^{m}; φ; θ; kG̃/I(kG̃)^{m}; φ̃]

such that Gφ = Q = G̃φ̃, and α is an isomorphism.

Suppose further that m = n + 2, and that α is induced by an automorphism of

Q. Then there exists an isomorphism G^{[n+1]}  G̃^{[n+1]} which induces α.

Proof. Note that for any group X, we have a canonical isomorphism

kX/I(kX)^{n+1} ≅ kX^{[n]}/I(kX^{[n]})^{n+1}.
Indeed, the kernel of the composition kX  kX^{[n]}  kX^{[n]}/I(kX^{[n]})^{n+1} is

I(kX)^{n+1} + I(kM_{n+1}(X))kX, and by (3.6), I(kM_{n+1}(X)) ⊆ I(kX)^{n+1}. The map φ above is the composition of the quotient map kG/I(kG)^{m} kG/I(kG)^{n+1} and this canonical isomorphism, and φ̃ is the composition of the similarly defined map kG̃/I(kG̃)^{m} kG̃^{[n]}/I(kG̃^{[n]})^{n+1} and an isomorphism arising from an isomorphism G̃^{[n]} ≅ Q.

The kernel of both φ and θφ̃ is I(kG)^{n+1}/I(kG)^{m}, so that there is an isomorphism α making the above diagram commutative.

Now suppose further that m = n + 2 and Qα = Q. By definition (3.4) of the Zassenhaus ideals, θ maps H_{n+1}(kG)/I(kG)^{n+2} onto H_{n+1}(kG̃)/I(kG̃)^{n+2}. By (3.7) and (3.6), these quotients are naturally isomorphic to M_{n+1}(G)/M_{n+2}(G) and M_{n+1}(G̃)/M_{n+2}(G̃), respectively. In particular, G^{[n+1]} and G̃^{[n+1]} have the same order. Choose a subspace C of kG̃ with I(kG̃)^{n+2} ≤ C ≤ I(kG̃)^{n+1} such that C/I(kG̃)^{n+2} is a complement to H_{n+1}(kG̃)/I(kG̃)^{n+2} in I(kG̃)^{n+1}/I(kG̃)^{n+2}. Note that C is an ideal of kG̃, and that φ̃ factors over kG̃/C. Identify the image of G in kG/I(kG)^{n+2} with G^{[n+1]}, and the image of G̃ in kG̃/I(kG̃)^{n+2} or kG̃/C with G̃^{[n+1]}. Then G^{[n+1]}θ embeds into kG̃/C since

(G^{[n+1]}θ − 1) ∩ I(kG̃)^{n+1}/I(kG̃)^{n+2} = H_{n+1}(kG̃)/I(kG̃)^{n+2}.

Also G^{[n+1]}θφ̃ = Q by assumption, and the pre-image of Q in (kG̃/C)^{×} is G̃^{[n+1]}. It follows that G^{[n+1]} maps onto G̃^{[n+1]} in kG̃/C, which yields the desired isomorphism. □

Röhl’s result is slightly stronger than the above proposition. In the end, we shall give another proof thereof, which in particular avoids induction on n, and shows that there is no need for replacing kF by a (completed) inverse limit of its quotients by powers of the augmentation ideal.

Note that by (3.7), the assumption M_{n+2}(G̃) = 1 in the following theorem is automatically fullfilled if kG ≅ kG̃.

Theorem 6.8 ([20, Theorem 3.1.2]). Let M_{n+2}(F) ≤ U ≤ M_{n+1}(F) for some n ≥ 1, and set G = F/U. Then kG/I(kG)^{n+2} ≅ kG̃/I(kG̃)^{n+2} for a group G̃ with M_{n+2}(G̃) = 1 implies G ≅ G̃.

Proof. Our assumption on G is that G^{[n]} = F^{[n]} = H and M_{n+2}(G) = 1.

Let θ : kG/I(kG)^{n+2} kG̃/I(kG̃)^{n+2} be an isomorphism, and let X denote either G or G̃. Note that since X is a p-group, I(kX)/I(kX)^{n+2} is the radical of kX/I(kX)^{n+2}.

By (3.7) and (3.6), the elementary abelian quotient M_{i}(X)/M_{i+1}(X) is isomorphic to the image of the Zassenhaus ideal H_{i}(kX) in kX/I(kX)^{i+1}, and by the definition (3.4) of the Zassenhaus ideals, these images are determined by the ring kX/I(kX)^{n+2} as long as i ≤ n+1. Therefore M_{i}(G)/M_{i+1}(G) and M_{i}(G̃)/M_{i+1}(G̃) are isomorphic for i ≤ n + 1. In particular, |G| = |G̃|. Moreover G̃^{[n]} ≅ H as any group X with M_{n+1}(X) = 1 and X/M_{2}(X) ≅ G/M_{2}(G) is a homomorphic image of H.

Thus by Lemma 6.7, we have a commutative diagram

[figure: kH/I(kH)^{n+1}; α; kH/I(kH)^{n+1}; kG/I(kG)^{n+2}; φ; θ; kG̃/I(kG̃)^{n+2}; φ̃]

with Gφ = H = G̃φ̃. By Lemma 6.4, we may assume, possibly after modifying φ̃ with an automorphism induced by an automorphism of H, that α is unipotent.

The kernel of the natural map kE kH/I(kH)^{n+1} is I(kE)^{n+1} + I(kV)kE, and by Lemma 6.3, we have V = M_{n+1}(E) and I(kV)kE ⊆ I(kE)^{n+1}. Thus we have an exact sequence 0 I(kE)^{n+1} kE kH/I(kH)^{n+1} 0 from which we obtain the exact sequence

[figure: 0; I(kE)^{n+1}/I(kE)^{n+2}; kE/I(kE)^{n+2}; τ; kH/I(kH)^{n+1}; 0; 0; J; ‖; A; ‖; τ; A/J; ‖; 0]

Let σ : kF/I(kF)^{n+2} kE/I(kE)^{n+2} be the map induced by the natural map kF kE = kF/L, and note that the kernel of σ is I(kL)kF + I(kF)^{n+2}/I(kF)^{n+2}. As σ is surjective on units, the universal property of F provides us with a commutative diagram

[figure: kF/I(kF)^{n+2}; σ; kE/I(kE)^{n+2}; τ; kH/I(kH)^{n+1}; kF/I(kF)^{n+2}; β; σ; kE/I(kE)^{n+2}; τ; kH/I(kH)^{n+1}; α]

where β is unipotent by (6.2). Let π : kF kF/I(kF)^{n+2} be the natural map. By commutativity, Rπβσ maps to 1 under τ, so Rπβσ is contained in 1 + I(kE)^{n+1}/I(kE)^{n+2}, an elementary abelian central p-group in the units of kE/I(kE)^{n+2}. As L = [F, R]R^{p}, it follows that Lπβσ = 1, which means that β stabilizes the kernel of σ. Hence there is an induced commutative diagram

[figure: 0; J; A; A/J; 0; 0; J; ‖; A; α̂; A/J; α; 0]

(α̂ induces the identity on J since α̂ is unipotent, too, and J = I(kE)^{n+1}/I(kE)^{n+2}).

Choose a subspace K of J which is a complement to the image of I(kM_{n+1}(E)) in J. Then K is an ideal of A, and the image of E in A/K is Ē = E/M_{n+2}(E). Moreover, J/K ≅ {M_{n+1}(E)/M_{n+2}(E) − 1} = {V̄ − 1} by Lemma 6.3, so that

[figure: 1; 1 + J/K; τ^{−1}(H)/K; H; 1; 1; V̄; ‖; Ē; ‖; H; ‖; 1]

Altogether, we obtain a commutative diagram of the form

[figure: J/K; ⊂; A/K; A/J]

[figure: J/K; ⊂; = = = =; A/K; A/J]
\frac{kG̃}{I(kG̃)^{n+2}}^{φ̃ }

[figure: \frac{kG}{I(kG)^{n+2}}; φ; θ]

The ideals of A/K contained in J/K are in one-to-one correspondence with the subgroups of Ē contained in V̄, with an ideal I corresponding to the subgroup 1 + I. Let I be an ideal of A/K contained in J/K. The commutative diagram above shows that the existence of one commutative diagram of the form

[figure: \frac{Ē}{1+I}; H; G; φ; ,; \frac{A/K}{I}; A/J; \frac{kG}{I(kG)^{n+2}}; φ; ,; \frac{A/K}{I}; A/J; \frac{kG̃}{I(kG̃)^{n+2}}; φ̃; or; \frac{Ē}{1+I}; H; G̃; φ̃]

implies the existence of all of them. By assumption, G ≅ Ē/Ū, and there is a commutative diagram

[figure: Ē/Ū; H; G; φ]

As we have seen, this implies the existence of a commutative diagram

[figure: 1; V̄ /Ū; Ē/Ū; H; 1; G̃; φ̃]

Since V̄ /Ū ≤ Φ(Ē/Ū), and φ̃ is surjective, the diagonal map is surjective, too, and as |G| = |G̃|, it follows that G ≅ Ē/Ū ≅ G̃, and the theorem is proved. □

Corollary 6.9. Let M_{n+2}(F) ≤ U ≤ M_{n+1}(F) for some n ≥ 1, and set G = F/U. Then Aut(kG) = Aut(G) · UAut(kG).

Proof. We have G^{[n]} = F^{[n]} = H and M_{n+2}(G) = 1. Suppose that α is a unipotent automorphism of kH/I(kH)^{n+1}. As in the proof of Theorem 6.8, we then obtain a unipotent lift α̂, as shown below.

[figure: 0; I(kE)^{n+1}/I(kE)^{n+2}; kE/I(kE)^{n+2}; kH/I(kH)^{n+1}; 0]

[figure: 0; I(kE)^{n+1}/I(kE)^{n+2}; ‖; kE/I(kE)^{n+2}; α̂; kH/I(kH)^{n+1}; α; 0]

Note that G = F/U = (F/L)/(U/L) = E/(U/L). The kernel of the natural map kE kG/I(kG)^{n+2} is I(kE)^{n+2} + I(kU/L)kE, and as U ≤ M_{n+1}(F), the kernel of the natural map kE/I(kE)^{n+2} kG/I(kG)^{n+2} is contained in I(kE)^{n+1}/I(kE)^{n+2}. It follows that α can be lifted to a unipotent automorphism of kG/I(kG)^{n+2}.

Now let θ ∈ Aut(kG), and denote the induced automorphism of kG/I(kG)^{n+2} also by θ. By Lemma 6.7, we have a commutative diagram

[figure: kH/I(kH)^{n+1}; α; kH/I(kH)^{n+1}; kG/I(kG)^{n+2}; θ; kG/I(kG)^{n+2}]

where α is an isomorphism, and the vertical maps are the natural ones. By Lemma 6.4, α is the product of a unipotent automorphism and a group automorphism of H. We have just seen that the unipotent automorphism can be lifted to a unipotent automorphism of kG/I(kG)^{n+2}, and therefore the group automorphism of H can be lifted to a group automorphism σ of G, by Lemma 6.7. It follows that θσ^{−1} is a unipotent automorphism of kG, and the proof is complete. □

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Universität Stuttgart, Fachbereich Mathematik, Institut für Geometrie und Topologie, Pfaffenwaldring 57, 70550 Stuttgart, Germany

   E-mail address: hertweck@mathematik.uni-stuttgart.de

Universität Hannover, Institut für Mathematik, Lehrgebiet Algebra und Zahlentheorie, Welfengarten 1, 30167 Hannover, Germany

   E-mail address: soriano@math.uni-hannover.de