CENTRAL p-GROUP EXTENSIONS: LAUDAL OBSTRUCTION SPACES REVISITED

MARTIN HERTWECK AND MARCOS SORIANO

Abstract. In a recent attack on the modular isomorphism problem, Borge and Laudal have introduced the concept of obstruction spaces. This paper contains an introduction and characterization of obstruction spaces for diagrams of central p-group extensions. We also relate obstruction spaces with certain universal short exact sequences derived from the lower p-central and Brauer– Jennings–Zassenhaus series of arbitrary finite p-groups. Finally, we discuss the connection of these results with the modular isomorphism problem.

1. Introduction

In this paper we deal with obstruction spaces for diagrams of groups, a concept that has been introduced in Borge and Laudal’s recent attack on the modular isomorphism problem (see [2, 3]).

Obstruction spaces appear quite naturally when one considers the problem of lifting homomorphisms of, say, groups and algebras (see Section 2). The idea underlying the concept of obstruction spaces and a first application of it can be traced back, at least, to the preprint [6]. But it should also be mentioned that the idea of obstruction spaces is implicitly contained in the work of Roggenkamp, Scott and Wursthorn — where algorithms for the computation of isomorphisms (if they exist) between modular group algebras are developed — and already plays a key role there (see [1, 9, 11]).

Starting from a lifting problem for group homomorphisms, we derive in Section 2 the key characterization of obstruction spaces for group diagrams of central extensions of elementary abelian p-groups, p any prime (see Proposition 2.2). Our main concern are group homomorphisms, but we extend in a straightforward manner — thus avoiding to repeat the same arguments with Hochschild cohomology replacing usual group cohomology — the result in Proposition 2.2 to certain diagrams of algebras (see Corollary 2.3).

In Section 2 we deliberately follow closely the path marked in [3, Section 2], although it should be mentioned that there are more direct approaches to prove existence of obstruction spaces using other elementary cohomological arguments.

The main idea underlying Borge and Laudal’s approach to the modular isomorphism problem is to try to connect obstruction spaces with the lower p-central and the Brauer–Zassenhaus–Jennings series of finite p-groups (see the final remarks following Proposition 3.2). We elaborate on this theme in Section 3, where we state a connection between obstruction spaces and certain explicitly given (and computable) universal short exact sequences derived from a free presentation of a given finite p-group G (see Propositions 3.1 and 3.2). Our approach — which is captured and easily recoverable from the key diagram (3.1) — is simple and direct, and avoids the use of any pro-p-group theory whatsoever. In addition, it allows a uniform treatment of the lower p-central and the Brauer–Jennings–Zassenhaus series and can, in fact, be extended to other central series with elementary abelian subfactors that might be of interest.

Date: January 5, 2005.

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

The first author acknowledges support by the Deutsche Forschungsgemeinschaft.

Keywords and Phrases. p-group, central series, modular isomorphism problem, cohomology.

2. Obstructions for lifting 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:

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

We shall refer to 𝔼_{φ} just as a diagram. Then, we are interested in “liftings” φ̂ of φ, i.e., we ask for commutative diagrams of the form

E/M \overline{}H

[figure: E; φ̂; H; ‖]

G^{φ }

where M is a normal subgroup of E contained in N. More precisely, we are interested in the subgroups M allowing a lifting. The main result of this section states that there exists a unique minimal normal subgroup M in N with this property provided 𝔼 is a central extension of an elementary abelian p-group.

Starting with a diagram 𝔼_{φ}, we can form the pullback along φ: Setting

P = {(e, g) | e ∈ E, g ∈ G with eσ = gφ} ≤ E × G,

we have a commutative diagram

[figure: 𝔼 :; 1; N; E; σ; H; 1; ℙ :; 1; N; ‖; P; G; φ; 1]

Observe that this way N, when abelian, becomes a G-module.

We note that there is a lifting

[figure: E; σ; H; G; φ; φ̂] (2.1) if and only if the sequence ℙ is split, i.e., if there is a commutative diagram

[figure: ℙ :; 1; N; P; G; 1; 1; N; ‖; N ⋊ G; G; ‖; 1]

Indeed, if we have a lifting φ̂, then

P = (N × 1) ⋊ {(gφ̂, g) | g ∈ G} ≅ N ⋊ G,

and ℙ is a split extension; the converse is also obvious.

From now on, we assume that N is abelian, and let obs 𝔼_{φ} be the cohomology element in H^{2}(G, N) corresponding to the equivalence class of the sequence ℙ (cf. [10, Chapter 5]). Then obs 𝔼_{φ} = 0 if and only if there is a lifting as displayed in (2.1).

An explicit description of obs 𝔼_{φ} is obtained as follows. Let π : Gφ → E be a section (with respect to σ), i.e., a map π : Gφ → E satisfying (π(gφ))σ = gφ for all g ∈ G. Then for any x, y ∈ G,

π(xφ) · π(yφ) = f(x, y) · π(xyφ) (2.2)

for an element f(x, y) ∈ N, thus defining a map f : G × G → N which is a factor set representing the cohomology class obs 𝔼_{φ}.

Assume that a morphism from 𝔼 to another exact sequence is given:

[figure: 𝔼̃_{φψ} :; 1; Ñ; Ẽ; H̃; 1]

[figure: 𝔼_{φ} :; 1; N; α; E; θ; σ; H; ψ; 1]

G^{φ }

Note that θ|_{N} : N → Ñ induces a map θ_{∗} : H^{2}(G, N) → H^{2}(G, Ñ). Applying θ to (2.2), one easily sees that obs 𝔼̃_{φψ} = θ_{∗}(obs 𝔼_{φ}). If the injections N → E and Ñ → Ẽ are inclusions, then

obs 𝔼̃_{φψ} = α_{∗}(obs 𝔼_{φ}). (2.3)

We now turn to central extensions of elementary abelian p-groups. Let k = 𝔽_{p}, and set H^{2}(G, k)^{∗} = Hom_{k}(H^{2}(G, k), k), the dual of H^{2}(G, k). We shall need the following (well-known) homological algebra result.

Lemma 2.1. Let G be a finite group, and let V be a finite dimensional vector space over k, considered as trivial G-module. Then there is a natural isomorphism

Φ_{V} : H^{2}(G, V) ⟶ Hom_{k}(H^{2}(G, k)^{∗}, V),

which in particular implies that for any subspace U of V, with natural factor map ρ : V → V/U, we have a commutative diagram

H^{2}(G, V) \overline{Φ_{V}} Hom_{k}(H^{2}(G, k)^{∗}, V)

[figure: H^{2}(G, V/U); ρ_{∗}; Φ_{V/U}; Hom_{k}(H^{2}(G, k)^{∗}, V/U); ρ ∘ −]

Proof. Let B = {v_{1}, …, v_{n}} be a basis of V, and let π_{i} : V → k denote the i-th ‘coordinate map’, sending ∑_{j} a_{j}v_{j} (a_{j} ∈ k) to a_{i}. Then we have an isomorphism

κ_{B} : H^{2}(G, V) ⟶ ⨁_{i=1}^{n} H^{2}(G, k), ζ ↦ (π_{i} ∘ ζ)_{1≤i≤n},

depending on the chosen basis B. Next we have an isomorphism (also depending on B)

λ_{B} : ⨁_{i=1}^{n} H^{2}(G, k) ⟶ Hom_{k}(H^{2}(G, k)^{∗}, V),

(ζ_{i})_{1≤i≤n} ↦ (φ ↦ ∑_{i=1}^{n} φ(ζ_{i})v_{i}).

The composition Φ_{V} = λ_{B} ∘ κ_{B}, however, does not depend on B. For let B′ = {v′_{1}, …, v′_{n}} be another basis of V, and write v_{i} = ∑_{j} a_{ij}v′_{j} with a_{ij} ∈ k. If κ_{B}(ζ) = (ζ_{i})_{1≤i≤n} for some ζ ∈ H^{2}(G, V), then κ_{B′}(ζ) = (∑_{i} a_{ij}ζ_{i})_{1≤j≤n}, so

(λ_{B′} ∘ κ_{B′})(ζ) = (φ ↦ ∑_{j=1}^{n} φ( ∑_{i=1}^{n} a_{ij}ζ_{i})v′_{j} = ∑_{j=1}^{n} ∑_{i=1}^{n} φ(ζ_{i})a_{ij}v′_{j} = ∑_{i=1}^{n} φ(ζ_{i})v_{i}).

To show naturality, let W be a second vector space over k with basis {w_{1}, …, w_{m}} and corresponding coordinate maps p_{j} : W → k, 1 ≤ j ≤ m. Let f : V → W be a k-linear map, and write f(v_{i}) = ∑_{j=1}^{m} f_{ij}w_{j} (all f_{ij} ∈ k). Using p_{j} ∘f = ∑_{i=1}^{n} f_{ij}π_{i}, we obtain for ζ ∈ H^{2}(G, V)

f ∘ Φ_{V} (ζ) = f ∘ (φ ↦ ∑_{i=1}^{n} φ(π_{i} ∘ ζ)v_{i})

= φ ↦ ∑_{j=1}^{m} ( ∑_{i=1}^{n} f_{ij}φ(π_{i} ∘ ζ))w_{j}

= φ ↦ ∑_{j=1}^{m} φ(p_{j} ∘ f ∘ ζ)w_{j}

= Φ_{W} (f ∘ ζ),

i.e., the isomorphisms are natural. □

Given a diagram

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

where V is an elementary abelian p-group in the center of E, we let obspace 𝔼_{φ} be the image of the map Φ_{V} (obs 𝔼_{φ}), where Φ_{V} is the natural isomorphism from Lemma 2.1:

obspace 𝔼_{φ} = im Φ_{V} (obs 𝔼_{φ}) ≤ V.

The importance of obspace 𝔼_{φ} is explained by the following lemma.

Proposition 2.2. Let 𝔼_{φ} be a diagram as above. Then obspace 𝔼_{φ} is the unique minimal subgroup U of V such that there is a lifting, as shown below:

[figure: 𝔼̄_{φ} :; 1; V/U; ⊂; E/U; H; 1]

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

G^{φ }

Proof. Set ζ = obs 𝔼_{φ}, and consider the commutative diagram from Lemma 2.1. Then

ρ_{∗}(ζ) = 0 ⟺ ((ρ ∘ −) ∘ Φ_{V})(ζ) = 0 ⟺ obspace 𝔼_{φ} ≤ U

which proves the lemma since ρ_{∗}(ζ) = obs 𝔼̄_{φ} by (2.3). □

We mention that the statement in Proposition 2.2 cannot be extended to noncentral extensions of abelian p-groups, as [4, 4.1 Example] in particular shows.

It should be remarked that lifting properties of algebra homomorphisms can be derived along the same lines using Hochschild cohomology. We only demonstrate that some results can be deduced directly from Proposition 2.2.

For this purpose, let A be a finite dimensional k-algebra (still k = 𝔽_{p}). We shall assume that A/rad(A) ≅ k (such an algebra might be called k-local). Furthermore, let J be a two-sided ideal of A such that rad(A)J = 0 = Jrad(A) and J^{p} = 0. Let G be a finite group, and let φ : kG → A/J be an algebra homomorphism. Then we ask for those ideals I of A, contained in J, for which there exists an algebra homomorphism φ̂ such that the following diagram commutes:

[figure: 0; J/I; A/I; A/J; 0; kG; φ; φ̂] (2.4)

Let A^{×} denote the group of units of A, and note that 1 + rad(A) ⊆ A^{×}. Observe that any additive subgroup of J is already an ideal of A, and that any subgroup U of 1 + J is a normal subgroup of A^{×}. Moreover, there is a 1-1 correspondence between ideals of A contained in J and normal subgroups of A^{×} contained in 1+J, with an ideal I corresponding to the subgroup 1 + I.

Let π : A → A/J denote the canonical homomorphism. Then we can compare the following lifting problems:

[figure: \frac{A}{I}; A/J; kG; φ; and; \frac{π^{−1}(Gφ)}{1+I}; Gφ; G; φ]

The lifting problem on the right hand side is solved by Proposition 2.2, since in the diagram

[figure: 𝔼_{φ} :; 1; 1 + J; π^{−1}(Gφ); Gφ; 1,; G; φ]

1 + J is a central elementary abelian p-subgroup (by the assumptions made on J). Note that if there exists a lifting for the left diagram (of algebras), then the image of G in A/I is contained in π^{−1}(Gφ)/I. Since we have a natural isomorphism (A/I)^{×} ≅ A^{×}/1 + I, it follows that a lifting for the left diagram (of algebras) exists if and only if a lifting for the right diagram (of groups) exists.

Thus we can state:

Corollary 2.3. With the notation as above, there exists a unique minimal ideal I of A, contained in J, such that there exists a lifting φ̂, as shown in (2.4). In fact, 1 + I = obspace 𝔼_{φ}. □

3. The essential point of the Laudal obstruction

We discuss an observation of Laudal [6], and a variation thereof, appearing in [2, 3], in a new way. Firstly, we work with free groups rather than with free pro-p groups (but this causes no real differences). Secondly, we have chosen to introduce the relevant spaces from the very beginning, then showing that they are what they ought to be (obstruction spaces in the sense of Proposition 2.2), thus obtaining a more accessible and unified presentation.

Let G = G_{1} ≥ G_{2} ≥ G_{3} ≥ … be a descending series of normal subgroups of the finite group G, and set G^{[n]} = G/G_{n+1}. We assume that G_{n}/G_{n+1} is an elementary abelian p-group, contained in the center of G^{[n]}. (The prime p is fixed.)

Let F be a free group on a set of generators of G, and choose a surjection π : F → G. Let K_{n} be the pre-image of G_{n+1} under the map π, so K_{n}π = G_{n+1}, and for n ≥ 1, we have induced exact sequences

[figure: 1; K_{n}; F; π_{n}; G^{[n]}; 1.]

Set L_{n+1} = ⟨[F, K_{n}], K_{n}^{p}⟩ ⊴ F. Then

L_{n+1}π = ⟨[G, G_{n+1}], G_{n+1}^{p}⟩ ≤ G_{n+2},

i.e., L_{n+1} is contained in K_{n+1}, the kernel of π_{n+1}. Also note that K_{n+1} ≤ K_{n}.

Note that K_{n} is of finite index in F. Therefore K_{n} is a free group of finite rank by the Nielsen–Schreier Theorem (see [8, 6.1.1]), which clearly implies that K_{n}/L_{n+1} is finite. Thus K_{n}/L_{n+1} is an elementary abelian p-group in the center of F/L_{n+1}.

Let φ_{n} : G → F/K_{n} be the homomorphism which when composed with the isomorphism F/K_{n} → G^{[n]} induced by π_{n} yields the natural map α_{n} : G → G^{[n]}.

Summarizing, we have the following commutative diagram, where the rows 𝔼_{n}, 𝔼̄_{n} are central extensions of elementary abelian p-groups, and the columns are exact.

(3.1)  [figure: 1  1; 𝔼̄_{n,α_{n}} :; 1; G_{n+1}/G_{n+2}; G^{[n+1]}; G^{[n]}; 1; 𝔼_{n,φ_{n}} :; 1; K_{n}/L_{n+1}; F/L_{n+1}; F/K_{n}; ≅; 1; K_{n+1}/L_{n+1}; {= \atop =}; K_{n+1}/L_{n+1}; G; φ_{n}; 1; 1]

Note that diagram 𝔼̄_{n,α_{n}} allows the lift α_{n+1} : G → G^{[n+1]}. Thus we may apply Proposition 2.2 to conclude that

obspace 𝔼_{n,φ_{n}} ≤ K_{n+1}/L_{n+1}.

We are interested in circumstances when equality holds. This is for a simple reason, already noticed in [6]. Note that the sequence 𝔼_{n} only depends on F and K_{n}, and that K_{1} may be very well derived from F alone. The question is whether there is a way of “computing” the kernels K_{j} in a way which does not make full use of the knowledge of G. Wishful thinking might suggest such a way, since Proposition 2.2 has Corollary 2.3 as an analogue on the algebra side, which might result in an inductive process to determine the middle term of 𝔼̄_{n} (more precisely, the kernel K_{n+1}), only from the knowledge of the group algebra kG. But this appears to be a rather difficult problem, which certainly can be attacked only with additional tools.

Assume from now on that G is a p-group, that F is free on a minimal generating set of G, and that G_{2} = [G, G]G^{p} (the Frattini subgroup of G). Then K_{1} = [F, F]F^{p} and K_{1}/L_{n+1} is the Frattini subgroup of F/L_{n+1}, by Burnside’s Basis Theorem (see [8, 5.3.2]).

Probably the main observation in Laudal [6] is the following proposition.

Proposition 3.1. Assume that G = G_{1}, and inductively, G_{n+1} = ⟨[G, G_{n}], G_{n}^{p}⟩ (that is, we consider the lower p-central series of G; then L_{n+1} = K_{n+2} for n ≥ 1). Then K_{n+1}/L_{n+1} = obspace 𝔼_{n,φ_{n}}.

Proof. Set U = obspace 𝔼_{n,φ_{n}} and W = K_{n+1}/L_{n+1}. We already know U ≤ W. Set V = K_{n}/L_{n+1}, E = F/L_{n+1} and H = F/K_{n}, so that

[figure: 𝔼_{n,φ_{n}} :; 1; V; E; H; 1; G; φ_{n}] (3.2)

Since K_{n} ≤ K_{1}, the remark preceding the lemma shows that V is contained in the Frattini subgroup of E, and consequently V/U is contained in the Frattini subgroup of E/U.

By Proposition 2.2, there is a homomorphism φ̂_{n} : G → E/U such that the following diagram commutes:

[figure: 1; V/U; E/U; H; 1; G; φ_{n}; φ̂_{n}] (3.3)

Note that since φ_{n} is surjective, the p-group E/U is generated by V/U and Gφ̂_{n}. Since elements of the Frattini subgroup are nongenerators (see [8, 5.2.12]), it follows that φ̂_{n} is surjective. Also G_{n+1}φ̂_{n} ≤ V/U since G_{n+1} is the kernel of φ_{n}, so

G_{n+2}φ̂_{n} = ⟨[G, G_{n+1}], G_{n+1}^{p}⟩φ̂_{n} ≤ ⟨[E/U, V/U], (V/U)^{p}⟩ = 1,

and it follows that E/U is a homomorphic image of G^{[n+1]}.

On the other hand, G^{[n+1]} ≅ E/W, and |E/W | ≤ |E/U|. It follows that |E/W | = |E/U|, and W = U. □

We remark that if G_{n+2} = 1, then G ≅ (F/L_{n+1})/obspace 𝔼_{n,φ_{n}}.

Note that the series considered in Proposition 3.1 is the fastest descending central series with elementary abelian factors. Also other central series with this property are of interest, and it should not be difficult to modify our approach to get corresponding conclusions. For the proof of Proposition 3.1 we essentially needed G_{n+2} to be contained in the kernel of the lift φ̂_{n}, and in order to keep this feature, we have to enlarge L_{n+1} suitably.

This will be explained in detail by means of the Brauer–Jennings–Zassenhaus series (see [7, p. 481]): M_{1}(G) = G, and for n ≥ 2

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

where ⌈n/p⌉ is the smallest integer ≥ n/p. Thus for the rest of the section, G_{n} = M_{n}(G). We keep the introduced notation, except that we now set

L_{n+1} = ⟨[F, K_{n}], K_{a_{n}}^{p}⟩, (3.4)

where

a_{n} = ⌈\frac{n+2}{p}⌉ − 1 ≤ ⌈\frac{n+1}{p}⌉ ≤ n. (3.5)

Hence L_{n+1} is defined in terms of F and K_{j} (j ≤ n) only, which, as already mentioned, might be seen as a key aspect of the Laudal obstruction. Again,

L_{n+1}π = ⟨[G, G_{n+1}], G_{a_{n}+1}^{p}⟩
= ⟨[G, G_{n+1}], G_{⌈(n+2)/p⌉}^{p}⟩
= G_{n+2},

i.e., L_{n+1} ⊆ K_{n+1}, so that we obtain the diagram (3.2). Note that by (3.4), K_{n}/L_{n+1} is a central elementary abelian p-subgroup of F/L_{n+1}, and with U = obspace 𝔼_{n,φ_{n}}, we have a commutative diagram (3.3), which can be written in the form

[figure: 1; K_{n}/M; F/M; F/K_{n}; 1; G; φ_{n}; φ̂_{n}]

for some M ⊴ F with L_{n+1} ≤ M ≤ K_{n}. Note that K_{n}/M is contained in the Frattini subgroup of F/M since K_{1} = [F, F]F^{p}.

We claim that G_{j+2}φ̂_{n} ≤ L_{j+1}M/M for j ≥ 1. Therefore, we set K_{0} = F, L_{0} = F and L_{1} = K_{1}, and prove the claim by induction on j, j ≥ −1. Obviously G_{1}φ̂_{n} ≤ F/M = L_{0}M/M, and if j ≥ 0, we obtain inductively, using (3.5), that

G_{j+2}φ̂_{n} = ⟨[G, G_{j+1}], G_{⌈(j+2)/p⌉}^{p}⟩φ̂_{n}
≤ [F, L_{j}]L_{a_{j}}^{p}M/M
≤ [F, K_{j}]K_{a_{j}}^{p}M/M
= L_{j+1}M/M.

It follows that G_{n+2}φ̂_{n} = 1, and we obtain as in the proof of Proposition 3.1:

Proposition 3.2. Assume that G = G_{1} ≥ G_{2} ≥ G_{3} ≥ … is the Brauer–Jennings– Zassenhaus series, and that the L_{n+1} are defined as in (3.4). Then K_{n+1}/L_{n+1} = obspace 𝔼_{n,φ_{n}}.

Again, we remark that if G_{n+2} = 1, then G ≅ (F/L_{n+1})/obspace 𝔼_{n,φ_{n}}.

Finally, we remark that we do not know whether M_{n+1}(F/L_{n+1}) = K_{n}/L_{n+1}. Note that M_{n+1}(F/L_{n+1}) is contained in K_{n}/L_{n+1}, which maps onto G_{n+1}/G_{n+2} in diagram (3.1), so K_{n}/L_{n+1} = M_{n+1}(F/L_{n+1})K_{n+1}/L_{n+1}, but why M_{n+1}(F/L_{n+1}) should lie over the obstruction space K_{n+1}/L_{n+1} appears to be unclear.

We briefly pick up the reason from [3, Section 4] which seemingly led Borge and Laudal to consider the Brauer–Jennings–Zassenhaus series in connection with the outlined obstruction theory. To this end, assume that M_{n+1}(F/L_{n+1}) = K_{n}/L_{n+1}. Then M_{n+2}(F/L_{n+1}) = 1. Indeed,

M_{n+2}(F/L_{n+1}) = [F, M_{n+1}(F)]M_{⌈(n+2)/p⌉}(F)^{p}L_{n+1}/L_{n+1}
= M_{⌈(n+2)/p⌉}(F)^{p}L_{n+1}/L_{n+1},

and M_{⌈(n+2)/p⌉}(F)π = M_{⌈(n+2)/p⌉}(G) ≤ G_{n+2}, so M_{⌈(n+2)/p⌉}(F) ≤ K_{n+1}. Consider the sequence 𝔼_{n} shown in (3.2) and set 𝔪 = I(kF/L_{n+1}). Then Jenning’s result [5] gives injections

\begin{matrix}0 & \overline{} & & 𝔪^{n+1}/𝔪^{n+2} & \overline{} & (kF/L_{n+1})/𝔪^{n+2} & \overline{\begin{matrix}κ \\ \end{matrix}} & (kF/L_{n+1})/𝔪^{n+1} & 0 \\ 1 & \overline{} & K_{n}/L_{n+1} & \overline{} & F/L_{n+1} & ^{^{∪} } & \overline{} & G^{[n]} & ^{^{∪} } & \overline{1} \end{matrix}

Switching to diagrams of groups, we certainly have

𝔻_{n,α_{n}} : 1 \overline{}1 + 𝔪^{n+1}/𝔪^{n+2} \overline{}κ^{−1}(G^{[n]}) \overline{\begin{matrix}κ \\  \end{matrix}}G^{[n]} 1

𝔼_{n,φ_{n}} : 1  \overline{}K_{n}/L_{n+1}^{^{∪} }  \overline{}F/L_{n+1}^{^{∪} }  \overline{}F/K_{n}^{≅ } 1

G^{φ_{n} }

If, by any chance, obspace 𝔻_{n,α_{n}} would essentially depend on kG alone, and, furthermore, we would have an inclusion obspace 𝔼_{n,φ_{n}} ⊆ obspace 𝔻_{n,α_{n}} (so that this is actually an equality), then F/K_{n+1} would be determined by Proposition 3.2, leading to some inductive process showing that the G^{[n]} can be recovered from the group algebra kG.

References

1. Frauke M. Bleher, Wolfgang Kimmerle, Klaus W. Roggenkamp, and Martin Wursthorn, Computational aspects of the isomorphism problem, Algorithmic algebra and number theory (Heidelberg, 1997), Springer, Berlin, 1999, pp. 313–329. MR 2000c:20007

2. Inger Christin Borge and Olav Arnfinn Laudal, The modular isomorphism problem, Univ. of Oslo, Dept. of Math., Preprint series in Pure Mathematics 2002, no. 19, ISBN 82-553-1357-5,

(http://www.math.uio.no/eprint/).

3. , The modular isomorphism problem, Univ. of Oslo, Dept. of Math., Preprint series in Pure Mathematics 2003, no. 25, ISBN 82-553-1393-1, (http://www.math.uio.no/eprint/).

4. Martin Hertweck, Contributions to the integral representation theory of groups, (http://

elib.uni-stuttgart.de/opus), 2004, Habilitationsschrift.

5. S. A. Jennings, The structure of the group ring of a p-group over a modular field, Trans. Amer. Math. Soc. 50 (1941), 175–185. MR 3,34f

6. O. A. Laudal, p-groups and Massey products, Preprint series 1975/76 No. 30, Mathematisk Institut, Aarhus Universitet.

7. Donald S. Passman, The algebraic structure of group rings, Wiley-Interscience [John Wiley & Sons], New York, 1977, Pure and Applied Mathematics. MR 81d:16001

8. Derek J. S. Robinson, A course in the theory of groups, second ed., Graduate Texts in Mathematics, vol. 80, Springer-Verlag, New York, 1996. MR 96f:20001

9. K. W. Roggenkamp and L. L. Scott, Automorphisms and nonabelian cohomology: an algorithm, Linear Algebra Appl. 192 (1993), 355–382, Computational linear algebra in algebraic and related problems (Essen, 1992). MR 94i:20010

10. Joseph J. Rotman, An introduction to homological algebra, Pure and Applied Mathematics, vol. 85, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1979. MR 80k:18001

11. Martin Wursthorn, Isomorphisms of modular group algebras: an algorithm and its application to groups of order 2^{6}, J. Symbolic Comput. 15 (1993), no. 2, 211–227. MR 94h:20008

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

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

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

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