Group Automorphism Action on Group Cohomology

Eric Jespers and Alexander Zimmermann^{∗}

1991 AMS Subject Classification: primary 20C05, 20J06; secondary 16G30

Abstract An automorphism α of a group G is called almost inner if α(g) is conjugate to g in G for any g ∈ G. Obviously, there exist p-groups G and non inner automorphisms of G inducing the identity on the mod p cohomology ring H^{∗}(G, IF_{p}). We pose the question if there are p-groups G and almost inner automorphisms which are not inner but which induce the identity on H^{∗}(G, IF_{p}). In the present paper we treat some examples to give evidence that there are no such groups G.

Introduction

Let G be a finite group and R be a commutative ring with trivial G-action. Then, any automorphism α of G induces by functoriality an automorphism of the cohomology ring H^{∗}(G, R). Jackowski and Marciniak proved [8] that in case α induces an inner automorphism of the group ring RG, then α^{∗} is the identity. Furthermore, they posed the following question ([8, Question 4.10] and [7]) for a p-group G: is any automorphism α of G necessarily inner whenever α induces the identity on H^{∗}(G, ℤ)? Note that for non p-groups a generalization of this question has a negative solution by an example of Hertweck [4]. Indeed, recently, in [4] Hertweck constructed a non p-group G and a non inner automorphism α of G which becomes inner in ℤG.

What might be the above Jackowski’s question for mod p cohomology? The Quillen stratification of the cohomology variety implies that an automorphism α of the p-group G which is the identity on the mod p cohomology ring fixes each conjugacy class of maximal elementary abelian subgroups. Obviously, there exist p-groups G and non inner automorphisms inducing the identity on H^{∗}(G, 𝔽_{p}). So, for mod p cohomology, it seems natural to restrict to almost inner group automorphisms α of a p-group G, that is, α(g) is conjugate to g for any g ∈ G. As mentioned by Jackowski and Marciniak [8] the group C_{8}×Aut(C_{8}) = C_{8}×(C_{2}×C_{2}) has an almost inner automorphism α that is not inner; however α^{∗} is not trivial. Here we denote by C_{n} the cyclic group of order n.

Let A be an abelian p-group acting on the cyclic p-group C_{p^{n}}. In this paper we prove that any almost inner automorphism of C_{p^{n}} ×A which induces the identity on H^{∗}(C_{p^{n}} ×A, 𝔽_{p}) is inner. Actually we show the following result.

∗This research was done while the authors collaborated at the ”Mathematisches Forschungsinstitut Oberwolfach” financed by the ”Research in Pairs” program of the ”Volkswagen Stiftung”. The first named author was also supported in part by NSERC-grant OGP0036631 and Fonds voor Wetenschappelijk Onderzoek, and the second named author got partial support from the CCCI.

Theorem 1 Let G be a semidirect product of a cyclic p-group by an abelian p-group. Let α be an almost inner but not inner automorphism of G. Then, α induces a non trivial automorphism of H^{3}(G, 𝔽_{p}).

In the final remark we mention that this result remains valid for the group G = C_{8} ×Q_{8} (and the field 𝔽_{2}). Here Q_{8} denotes the quaternion group of order 8. Note that this group has been used in [9] to construct a group H with a non inner automorphism but which induces an inner automorphism on the group ring SH over some semilocal ring of algebraic integers (thus a semilocal version of Hertweck’s example).

The paper is organized as follows. In section 1 and 2 we introduce the notation and recall some known facts. In section 3 we restrict our group theoretical setup, and in section 4 we do the cohomology constructions.

1 Notations

Let G be a finite group and let α be an automorphism of G. For any ℤG-module M we denote by ^{α}M the ℤG-module with the same additive structure as M but g ∈ G acts on αM as α(g) acts on M. Let R be a commutative ring and view R as an RG-module via the trivial action. Then, α acts on the cohomology ring H^{∗}(G, R) via the following construction (see e.g. [2, page 80]). Take a projective resolution P of ℤ as ℤG-module. Then, P is a chain complex with homology ℤ in degree 0 and exact in all the other degrees. The complex αP has homology ^{α}ℤ ≃ ℤ in degree 0. Hence, there is a chain map

τ : P ⟶ ^{α}P

with

τ = ⨁_{n∈ℕ}τ_{n},

where

τ_{n} : P_{n} ⟶ ^{α}P_{n}

is the mapping between the homogenous components. Any two such mappings τ differ by a homotopy. Applying Hom_{ℤG}(−, R) to P and to ^{α}P the mapping

τ : P ⟶ ^{α}P

induces a cochain map

τ^{∗} := Hom(τ, R) : Hom_{ℤG}( ^{α}P, R) ⟶ Hom_{ℤG}(P, R)

Moreover, the identity map gives an identification

Hom_{ℤG}( ^{α}P, R) ≃ Hom_{ℤG}(P, ^{α^{−1}}R) ≃ Hom_{ℤG}(P, R)

Since τ^{∗} is a cochain map, it induces a mapping on the cohomology of the complexes:

τ^{∗} : H^{∗}(Hom_{ℤG}( ^{α}P, R)) ⟶ H^{∗}(Hom_{ℤG}(P, R))

The map

τ^{n} : H^{n}(Hom_{ℤG}( ^{α}P, R)) ≃ H^{n}(G, R) ⟶ H^{n}(G, R) ≃ H^{n}(Hom_{ℤG}(P, R))

is the mapping induced by α on the cohomology.

Observe that the map τ is defined as soon as ^{α}R ≃ R. This in turn is true not only for trivial G-modules R.

We discuss the possible form of τ_{n}.

Lemma 1 Let P be a free RG-module of rank n and let α be an automorphism of RG. Then, choosing a basis of P, any isomorphism ^{α}P ⟶^{φ} P can be decomposed as φ(x) = α^{−1}(x)M for a n × n-matrix M with coefficients in RG.

Proof. Choose a basis for P. We then obtain an RG-isomorphism α_{0} : P ⟶ ^{α}P by applying α to each of the components (with respect to this basis). The automorphism α_{0} ∘ φ is determined by a n × n-matrix N with coefficients in RG; that is, α_{0}(φ(x)) = xN. So the result follows.

2 Constructing projective resolutions

For the reader’s convenience, we recall a method due to C. T. C. Wall [10] for constructing projective resolutions for group extensions.

Let

1 ⟶ N ⟶ G ⟶ H ⟶ 1

be an exact sequence of finite groups and let B be a free resolution of the ℤN-module ℤ. Let C be a free resolution for the ℤH-module ℤ. Then, ℤG ⊗_{ℤN} B is a free resolution for the ℤG-module

ℤG ⊗_{ℤN} ℤ ≃ ℤH.

Take a free resolution C = (C_{n}, d_{1}) of ℤH. Then, each of the homogeneous components C_{n} of C is a free ℤH-module and can be replaced by the above argument by a direct sum of the projective resolution ℤG ⊗_{ℤN} B. The corresponding identification map is the augmentation and will be denoted by ϵ as well. Call the k−th homogeneous component of the free resolution of the n−th homogeneous component of C the ℤG-module A_{n,k}. Let d_{0} denote the differential of each of the free resolutions of C_{s}. Since B is exact in non zero degrees, there exist ℤG linear maps

d_{1} = (d_{1})_{n,m} : A_{n,m} → A_{n−1,m}

so that d_{1}d_{0} + d_{0}d_{1} = 0. In other words, there is a chain map

d_{1} : (ℤG ⊗_{ℤN} B)^{dim_{ℤH}C_{s}} ⟶ (ℤG ⊗_{ℤN} B)^{dim_{ℤH}C_{s−1}}

which lifts the differential d_{1} of C; but one takes alternate signs for differentials resolving C_{s} and C_{s+1}. Of course, d_{1} is not necessarily a differential again. Nevertheless, we have the following statement.

Proposition 2 (C. T. C. Wall [10]) There exists ℤG-linear endomorphisms d_{r} of bidegree (−r, r − 1), that is, there are ℤG-linear maps d_{r} : A_{n,m} → A_{n+r−1,m−r} (r ≥ 1, m ≥ r), such that

d_{1}ϵ = ϵd_{1} and ∑_{i=0}^{k} d_{i}d_{k−i} = 0

for each k. Moreover, (⨁_{k=1}^{n} A_{k,n−k}, ∑_{k=0}^{n} d_{k}) is a free resolution of the ℤG-module ℤ. Furthermore, any set of d_{r} satisfying the above properties gives rise to a free resolution of the ℤG-module ℤ.

Remark 3 The proof of proposition 2 is done inductively. That is, if one has found d_{k} which satisfies ∑_{i=0}^{k} d_{i}d_{k−i} = 0, then there are d_{n} with d_{0}, d_{1}, …, d_{k} already determined.

3 Almost inner automorphisms and metabelian groups

We shall discuss p-groups G which have a cyclic normal subgroup N = C_{p^{n}} and an abelian complement Ab. That is, we get a split extension of finite groups

1 ⟶ N ⟶ G ⟶ Ab ⟶ 1 .

As is well known (cf e.g. [6, Satz I.13.9]), the group of automorphisms of the cyclic p-group C_{p^{n}} has a Sylow p-subgroup which is cyclic in case p is odd and is isomorphic to C_{2} × C_{2^{n−2}} in case p = 2 (and n ≥ 2).

If p is odd,

G ≃ (C_{p^{n}} ×C_{p^{k}}) × Ãb

for some k and Ãb an abelian group. Now, if α is an almost inner automorphism of G, then any element x ∈ G is conjugate to its image α(x). So, since Ãb is abelian, it follows that α acts as the identity on Ãb and α induces an automorphism on (C_{p^{n}} ×C_{p^{k}}).

Lemma 4 Let G be a split metacyclic group. Then, any almost inner automorphism of G is inner.

Proof. Let α be an almost inner automorphism on G = C ×A, with C and A cyclic groups. Suppose Ab = ⟨a⟩. As C is normal in G, α induces an automorphism on C. Since C is cyclic, we may use an inner automorphism of G to modify α in such a way that α is the identity on C. As G/C is abelian, we get α(a) = ac = xax^{−1} for some x ∈ G and c ∈ C. Again because Ab is abelian, we may also assume that x ∈ C. It follows that α is conjugation by x.

Because of Lemma 4 and the preceding discussion we get at once the following application.

Corollary 5 If p is odd and if G is a split extension of a cyclic p-group and an abelian p-group, then any almost inner automorphism of G is inner.

Let now p = 2 (n ≥ 2) and thus Aut(N) = C_{2} × C_{2^{n−2}}. Let α be an almost inner automorphism of G. If the image of Ab → Aut(N) is cyclic, then we again may apply Lemma 4.

Lemma 6 Let G = C_{2^{n}} ×Ab for an abelian group Ab and let α be an almost inner automorphism of G. If the induced homomorphism Ab ⟶ Aut(C_{2^{n}}) has a cyclic image, then any almost inner automorphism of G is inner.

Proof. In this case,

G ≃ (C_{2^{n}} ×C_{2^{k}}) × Ãb

for a certain k and an abelian group Ãb. Therefore, any almost inner automorphism α induces an almost inner automorphism on each of the factors. Since Ãb is abelian and since C_{2^{n}} ×C_{2^{k}} is split metacyclic, the induced automorphisms on the factors are inner. Thus α is inner.

We now shall discuss the remaining case where in fact almost inner automorphisms can occur. So p = 2, G = C_{2^{n}} × Ab for an abelian group Ab, and the induced mapping φ of Ab to the automorphism group Aut(C_{2^{n}}) = C_{2^{n−2}} ×C_{2} does not have a cyclic image. Thus φ(Ab) = C_{2^{m}} × C_{2} with m ≤ n − 2. Here the factor group C_{2} is generated by the automorphism that maps a to a^{−1}, where a is the generator of C_{2^{n}}. It follows that Ab = ⟨b⟩ × ⟨c⟩ × Ãb, so that bab^{−1} = a^{−1} and c induces the automorphism of order 2^{m} on C_{2^{n}}. Any almost inner automorphism α of G induces the identity on Ãb and an almost inner automorphism on C_{2^{n}} ×(⟨b⟩ × ⟨c⟩). So we shall now discuss the action of α on the latter group.

Lemma 7 Let G = C_{2^{n}} × (C_{2^{k}} × C_{2^{l}}). Suppose C_{2^{n}} = ⟨a⟩, C_{2^{k}} = ⟨b⟩, C_{2^{l}} = ⟨c⟩ and bab^{−1} = a^{−1}. Then G has an almost inner automorphism α_{0} so that α_{0}(c) = a^{2^{n−1}}c and α_{0} acts trivially on ⟨a, b⟩. Furthermore, α_{0} is not inner and for any almost inner automorphism α of G there exists an inner automorphism γ so that αγ either equals α_{0} or the identity.

Proof. Let α be an almost inner automorphism of G. Because of Lemma 4, we may modify α by an inner automorphism so that

α(a) = a and α(b) = b.

Hence, α induces an almost inner automorphism on the abelian group G/⟨a, b⟩

α(c) = a^{s} · c

for some s. As bc = cb, we get b · a^{s} · c = α(b)α(c) = α(c) · α(b) = a^{s} · c · b . Therefore, a^{−s} = a^{s} which implies s ∈ {0, 2^{n−1}}.

Clearly α_{0} is an automorphism. We now prove that it is almost inner. Write cac^{−1} = a^{v} and v = 2^{r}v′ + 1 with 0 ≤ r < 2^{n} and (v′, 2) = 1. Let q = 2^{n−1−r}. Then q(v − 1) = 2^{n−1}v′. So it follows that

a^{q}ca^{−q}c^{−1} = a^{q}a^{−vq} = a^{q(1−v)} = a^{2^{n−1}}

and thus α_{0} = a^{2^{n−1}}c = a^{q}ca^{−q}. Hence, for any integers x, y,

α_{0}(a^{x}b^{2y}c) = a^{x}b^{2y}a^{2^{n−1}}c = a^{x}b^{2y}a^{q}ca^{−q} = a^{q}(a^{x}b^{2y}c)a^{−q}.

Now to deal with odd exponents of b, write v + 1 = 2^{r′}v′′ with 1 ≤ r′ ≤ n and (v′′, 2) = 1. It follows that

α_{0}(a^{x}b^{2y+1}c) = a^{x}b^{2y+1}a^{2^{n−1}}c
= a^{x}b^{2y+1}(a^{2^{n−1}})^{v′′}c
= a^{x}b^{2y+1}a^{−2^{n−1−r′}2^{r′}v′′}c
= a^{x}b^{2y+1}a^{−2^{n−1−r′}(1+v)}c
= a^{2^{n−1−r′}} a^{x}b^{2y+1}c(c^{−1}a^{−v2^{n−1−r′}} c)
= a^{2^{n−1−r′}} (a^{x}b^{2y+1}c)a^{−2^{n−1−r′}}

Hence this proves that α_{0} is indeed almost inner.

Finally, suppose α_{0} is inner. Since ⟨b, c⟩ is abelian, there exists x ∈ N so that

a^{x}ca^{−x} = a^{2^{n−1}}c

and

a^{x}bca^{−x} = a^{2^{n−1}}bc.

Therefore a^{x}ca^{−x} = b^{−1}a^{x}bca^{−x} and thus a^{x}ca^{−x} = a^{−x}ca^{−x}. So a^{2x} = 1 and thus 2^{n−1}|x. But then a^{x} is central and this yields a contradiction.

4 Computing H^{∗}(G, 𝔽_{p})

In this section we will prove Theorem 1. So we consider a p-group G which is a semidirect product of a cyclic p-group by an abelian p-group. In order that G has an almost inner automorphism α that is not inner, the results in Section 3 yield that p = 2 and

G ≃ (C_{2^{n}} ×(C_{2^{k}} × C_{2^{l}})) × Ãb,

for some abelian group Ãb. Moreover, up to inner automorphisms α is unique.

4.1 H^{∗}(G, IF_{p}) as vector space

We are only interested in cohomology with values in 𝔽_{p}. By the Künneth formula,

H^{n}(G, 𝔽_{p}) = ⨁_{r=0}^{n} H^{r}(Ãb, 𝔽_{p}) ⊗_{ℤ} H^{n−r}(C_{p^{n}} ×(C_{p^{k}} × C_{p^{l}}), 𝔽_{p})).

Note that for integral cohomology the situation would be more difficult. Since our automorphism is the identity on Ãb, we may trace the action of α by the knowledge of the action of α on H^{∗}(C_{p^{n}} ×(C_{p^{k}} × C_{p^{l}}), 𝔽_{p}).

So, we may assume

G = C_{2^{n}} ×(C_{2^{k}} × C_{2^{l}})

and we use the same notation as in the statement of Lemma 7. Let m be a positive integer so that c^{−1}ac = a^{m}. For an element g ∈ G of order n we denote Δ_{g} := ∑_{i=0}^{n−1} g^{i}. Put, A := a − 1, B := b − 1 and C := c − 1.

We shall have to give an explicit projective resolution of ℤG-modules of the trivial module ℤ. For this we shall use Proposition 2 several times. We start with the standard projective resolution of the trivial ℤC_{2^{n}}-module ℤ:

ℤ ⟵^{ϵ} ℤC_{2^{n}} ⟵^{A} ℤC_{2^{n}} ⟵^{Δ_{a}} ℤC_{2^{n}} ⟵^{A} ℤC_{2^{n}} ⟵^{Δ_{a}} ℤC_{2^{n}} ⟵^{A} ℤC_{2^{n}} ⟵^{Δ_{a}} …

A complex giving a free resolution of the trivial ℤG/C_{2^{n}}-module ℤ is obtained from the total complex of the tensor product of the standard projective resolutions, as ℤC_{2^{k}} and ℤC_{2^{l}} respectively, of the trivial module ℤ.

Set Λ := ℤC_{2^{k}} × ℤC_{2^{l}}. Using Proposition 2 we get a projective resolution

0 ⟵ ℤ ⟵ Λ ⟵ Λ^{2} ⟵ Λ^{3} ⟵ Λ^{4} ⟵ Λ^{5} ⟵

The differentials are

d^{1} = \binom{C}{B}; d^{2} := ( \begin{matrix}Δ_{c} & 0 \\ −B & C \\ 0 & Δ_{b} \end{matrix} ); d^{3} := ( \begin{matrix}C & 0 & 0 \\ B & Δ_{c} & 0 \\ 0 & −Δ_{b} & C \\ 0 & 0 & B \end{matrix} ); d^{4} := ( \begin{matrix}Δ_{c} & 0 & 0 & 0 \\ −B & C & 0 & 0 \\ 0 & Δ_{b} & Δ_{c} & 0 \\ 0 & 0 & −B & C \\ 0 & 0 & 0 & Δ_{b} \end{matrix} ); …

Let Γ := ℤG. To compute a complex giving a free resolution of the trivial ℤG-module ℤ we consider the total complex of the tensor product of the standard resolution of ℤC_{2^{n}}-modules and the resoution obtained for Λ-modules. So we obtain the diagram:

Γ_{0,0} ← Γ_{1,0}^{2} ← Γ_{2,0}^{3} ← Γ_{3,0}^{4} ← Γ_{4,0}^{5} ← Γ_{5,0}^{6}
↑ ↑ ↑ ↑ ↑ ↑
Γ_{0,1} ← Γ_{1,1}^{2} ← Γ_{2,1}^{3} ← Γ_{3,1}^{4} ← Γ_{4,1}^{5} ← Γ_{5,1}^{6}
↑ ↑ ↑ ↑ ↑ ↑
Γ_{0,2} ← Γ_{1,2}^{2} ← Γ_{2,2}^{3} ← Γ_{3,2}^{4} ← Γ_{4,2}^{5} ← Γ_{5,2}^{6}
↑ ↑ ↑ ↑ ↑ ↑
Γ_{0,3} ← Γ_{1,3}^{2} ← Γ_{2,3}^{3} ← Γ_{3,3}^{4} ← Γ_{4,3}^{5} ← Γ_{5,3}^{6}
↑ ↑ ↑ ↑ ↑ ↑
Γ_{0,4} ← Γ_{1,4}^{2} ← Γ_{2,4}^{3} ← Γ_{3,4}^{4} ← Γ_{4,4}^{5} ← Γ_{5,4}^{6}
↑ ↑ ↑ ↑ ↑ ↑
Γ_{0,5} ← Γ_{1,5}^{2} ← Γ_{2,5}^{3} ← Γ_{3,5}^{4} ← Γ_{4,5}^{5} ← Γ_{5,5}^{6}
↑ ↑ ↑ ↑ ↑ ↑
Γ_{0,6} ← Γ_{1,6}^{2} ← Γ_{2,6}^{3} ← Γ_{3,6}^{4} ← Γ_{4,6}^{5} ← Γ_{5,6}^{6}

The vertical differentials

d_{0} : Γ_{i,j}^{i+1} ⟶ Γ_{i,j−1}^{i+1}

are multiplication by A if j is odd and multiplication by Δ_{a} if j is even.

To define d_{1} we proceed in several steps. The morphisms

d_{1} : Γ_{i,0}^{i+1} ⟶ Γ_{i−1,0}^{i}

are determined by the same matrix as the morphisms

d_{1} : Λ_{i,0}^{i+1} ⟶ Λ_{i−1,0}^{i} .

The morphisms

d_{1} : Γ_{i,1}^{i+1} ⟶ Γ_{i−1,1}^{i}

and

d_{1} : Γ_{i,2}^{i+1} ⟶ Γ_{i−1,2}^{i}

are defined by matrices obtained from the matrices in degree (∗, 0) by replacing c by c∇_{a}^{m} and b by b∇_{a}^{2^{n}−1}. Continuing this way, the morphisms

d_{1} : Γ_{i,2f−1}^{i+1} ⟶ Γ_{i−1,2f−1}^{i}

and

d_{1} : Γ_{i,2f}^{i+1} ⟶ Γ_{i−1,2f}^{i}

are defined by matrices obtained from the matrices in degree (∗, 0) by replacing c by c(∇_{a}^{m})^{f} and b by b(∇_{a}^{2^{n}−1})^{f}. Now note that

Ab = b(a − 1)∇_{a}^{2^{n}−1} and Ac = c(a − 1)∇_{a}^{m}

and thus

A (c(∇_{a}^{m})^{f−1} − 1) = (c(∇_{a}^{m})^{f} − 1) A and A (b(∇_{a}^{2^{n}−1})^{f−1} − 1) = (b(∇_{a}^{2^{n}−1})^{f} − 1) A.

Since also Δ_{a} is central in ℤG and one then easily verifies that the diagram is commutative, i.e., d_{0}d_{1} = d_{1}d_{0}. The diagram does not have exact lines, however. Nevertheless, since the extension for G splits, in degree (∗, 0) the sequence is exact. Next, define

0 = d_{2} : Γ_{i,2f}^{i+1} ⟶ Γ_{i−2,2f+1}^{i−1}

and

d_{2} : Γ_{i,2f−1}^{i+1} ⟶ Γ_{i−2,2f}^{i−1}

is determined by the (i + 1) × (i − 1)-matrix

d_{2} := − ( \begin{matrix}λ_{f} & 0 & … & … & … & … & 0 \\ 0 & λ_{f} & 0 & & & & ⋮ \\ κ_{f} & 0 & λ_{f} & 0 & & & ⋮ \\ 0 & κ_{f} & 0 & λ_{f} & 0 & & ⋮ \\ ⋮ & ... & ... & ... & ... & ... & ⋮ \\ ⋮ & & ... & ... & ... & ... & 0 \\ ⋮ & & & 0 & κ_{f} & 0 & λ_{f} \\ ⋮ & & & & 0 & κ_{f} & 0 \\ 0 & … & … & … & … & 0 & κ_{f} \end{matrix} )

with

κ_{f} := \frac{(2^{n} − 1)^{f·2^{k}} − 1}{2^{n}} and λ_{f} := \frac{m^{f·2^{l}} − 1}{2^{n}},

That the latter numbers are integers follows from the following calculations.

Since c∇_{a}^{m}A = Ac we get

((c∇_{a}^{m})^{f})^{2^{l}} · A = Ac^{2^{l}} = A

and hence ((c∇_{a}^{m})^{f})^{2^{l}} − 1 = yΔ_{a} for some y ∈ ℤ < b, c >. Putting a = 1 one obtains

(cm)^{f2^{l}} − 1 = m^{f2^{l}} − 1 = y2^{n}.

This gives

((c∇_{a}^{m})^{f}) 2^{l} − 1 = \frac{m^{f2^{l}} − 1}{2^{n}}Δ_{a} .

Similarly, as ∇_{a}^{2^{n}−1} = Δ_{a} − a^{−1},

(b∇_{a}^{2^{n}−1})^{2^{k}} − 1 = b^{2^{k}} ((Δ_{a} − a)(Δ_{a} − a^{−1}))^{2^{k−1}} − 1
= ((2^{n} − 2)Δ_{a} + 1)^{2^{k−1}} − 1
= ((Δ_{a} − 1)(Δ_{a} − 1))^{2^{k−1}} − 1
= (Δ_{a} − 1)^{2^{k}} − 1
= ∑_{i=1}^{2^{k}}(−1)^{2^{k}−i}\binom{2^{k}}{i}Δ_{a}^{i}

= ∑_{i=1}^{2^{k}}(−1)^{2^{k}−i}\binom{2^{k}}{i}2^{n(i−1)}Δ_{a}

= 2^{−n} (∑_{i=1}^{2^{k}}(−1)^{2^{k}−i}\binom{2^{k}}{i}(2^{n})^{i}) Δ_{a}

= (\frac{(2^{n} − 1)^{2^{k}} − 1}{2^{n}}) Δ_{a}

Thus (bΔ_{a}^{2^{n}−1})^{2^{k}} A = A. It follows that (bΔ_{a}^{2^{n}−1})^{2^{k}} − 1 ∈ ℤ⟨b, c⟩Δ_{a}. So we obtain for any f,

(b(∇_{a}^{2^{n}−1})^{f})^{2^{k}} − 1 = \frac{(2^{n} − 1)^{f·2^{k}} − 1}{2^{n}}Δ_{a}

Further one easily verifies that

c (∇_{a}^{m})^{f} b (∇_{a}^{2^{n}−1})^{f} = cb(mΔ_{a} − a^{−1}∇_{a^{−1}}^{m})
= cb(mΔ_{a} − (a^{−1} + a^{−2} + ⋯ + a^{−m})
= b (∇_{a}^{2^{n}−1})^{f} c (∇_{a}^{m})^{f}

Making use of all this information one now verifies that

d_{2}d_{0} + d_{1}d_{1} + d_{0}d_{2} = d_{1}d_{1} + d_{0}d_{2} = 0

in odd rows and

d_{2}d_{0} + d_{1}d_{1} + d_{0}d_{2} = d_{1}d_{1} + d_{2}d_{0} = 0

in even rows. So we found the first two terms of the spectral sequence leading to Wall’s criterion (Proposition 2) for a projective resolution.

Since the matrix for the differentials d^{k} in ℤ < b, c > have nonzero entries only in the diagonal and the lower diagonal, and because the matrix defining d_{2} never has two consecutive non zero entries, one observes by elementary computation that one can pose d_{3} = 0.

Since d_{2}d_{2} = 0 the defining equation for d_{4} degenerates to d_{0}d_{4} + d_{4}d_{0} = 0 and here d_{4} = 0 is possible. Consequently, we can take d_{r} = 0 for any r ≥ 3. Hence,

d := d_{0} + (−1)^{i}d_{1} + d_{2}

already is a differential (here the (−1)^{i} means that one applies d_{1} on even columns and one applies −d_{1} on odd columns).

Lemma 8 The differential d as defined above yields a projective resolution of the trivial ℤG-module ℤ.

Proof. This is an immediate consequence of the above discussion and the theorem of C. T. C. Wall.

Corollary 9 The spectral sequence provided by the bigraded module used in C.T.C. Wall’s construction for this group degenerates in degree 3.

We write down the first terms of this resolution explicitly.

Γ ← Γ^{2} ⊕ Γ ← Γ^{3} ⊕ Γ^{2} ⊕ Γ ← Γ^{4} ⊕ Γ^{3} ⊕ Γ^{2} ⊕ Γ ← Γ^{5} ⊕ Γ^{4} ⊕ Γ^{3} ⊕ Γ^{2} ⊕ Γ ← …

the differentials are

d_{1} := ( \begin{matrix}C \\ B \\ A \end{matrix} ); d_{2} := ( \begin{matrix}Δ_{c} & 0 & 0 \\ −B & C & 0 \\ 0 & Δ_{b} & 0 \\ A & 0 & 1 − c∇_{a}^{m} \\ 0 & A & 1 − b∇_{a}^{2^{n}−1} \\ 0 & 0 & Δ_{a} \end{matrix} );

d_{3} := ( \begin{matrix}C & 0 & 0 & 0 & 0 & 0 \\ B & Δ_{c} & 0 & 0 & 0 & 0 \\ 0 & −Δ_{b} & C & 0 & 0 & 0 \\ 0 & 0 & B & 0 & 0 & 0 \\ A & 0 & 0 & −Δ_{c∇_{a}^{m}} & 0 & −λ_{1} \\ 0 & A & 0 & b∇_{a}^{2^{n}−1} − 1 & 1 − c∇_{a}^{m} & 0 \\ 0 & 0 & A & 0 & −Δ_{b∇_{a}^{2^{n}−1}} & −κ_{1} \\ 0 & 0 & 0 & Δ_{a} & 0 & c∇_{a}^{m} − 1 \\ 0 & 0 & 0 & 0 & Δ_{a} & b∇_{a}^{2^{n}−1} − 1 \\ 0 & 0 & 0 & 0 & 0 & A \end{matrix} );

d_{4} :=

Δ_{c} 0  0  0  0  0  0  0  0  0
−B C 0  0  0  0  0  0  0  0
0 Δ_{b} Δ_{c} 0  0  0  0  0  0  0
0  0 −B C  0  0  0  0  0  0
0  0  0 Δ_{b}  0  0  0  0  0  0
A 0  0  0  1 − c∇_{a}^{m}  0  0  −λ_{1}  0  0
0 A 0  0 1 − b∇_{a}^{2^{n}−1} −Δ_{c∇_{a}^{m}}  0  0  −λ_{1}  0
0  0 A 0  0  Δ_{b∇_{a}^{2^{n}−1}} 1 − c∇_{a}^{m}  −κ_{1}  0  0
0  0  0 A  0  0  1 − b∇_{a}^{2^{n}−1}  0  −κ_{1}  0
0  0  0  0  Δ_{a}  0  0  Δ_{c∇_{a}^{m}}  0  0
0  0  0  0  0  Δ_{a}  0  1 − b∇_{a}^{2^{n}−1} c∇_{a}^{m} − 1  0
0  0  0  0  0  0  Δ_{a}  0  Δ_{b∇_{a}^{2^{n}−1}}  0
0  0  0  0  0  0  0  A  0  1 − c(∇_{a}^{m})^{2}
0  0  0  0  0  0  0  0  A  1 − b(∇_{a}^{2^{n}−1})^{2}
0  0  0  0  0  0  0  0  0  Δ_{a}

4.2 The action of the automorphism

We shall have to define homomorphisms

τ_{k} : ⊕_{j=1}^{k+1}Γ^{j} ⟶ ^{α}(⊕_{j=1}^{k−1}Γ^{j})

and then τ_{k} defines the action of α_{0} on the cohomology in degree k. Because of Lemma 1 we shall define matrices M_{k} such that τ_{k}(x) = α(x) · M_{k}.

For the moment we put ∇ := ∇_{a}^{2^{n−1}} and ∇^{x} := ∇_{a}^{x}. It is clear that we may put M_{0} = 1. Now, for k ≥ 1 the matrix k has to be such that

α_{0}(d_{k}) = M_{k}d_{k}.

It is easy to see that

M_{1} = ( \begin{matrix}a^{2^{n−1}} & 0 & ∇ \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )

and

M_{2} := ( \begin{matrix}1 & 0 & 0 & Δ_{c^{2}}∇ & 0 & \frac{1+m}{2}cΔ_{c^{2}} \\ 0 & a^{2^{n−1}} & 0 & 0 & ∇ & (2^{n−1} − 1)b \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & a^{2^{n−1}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} )

satisfy the desired equalities.

The matrix M_{3} can be taken as the following.

( \begin{matrix}a^{2^{n−1}} & 0 & 0 & 0 & ∇ & 0 & 0 & \frac{1}{2}(c − 1)Δ_{c^{2}} + Δ_{cm} & 0 & σ \\ 0 & 1 & 0 & 0 & 0 & Δ_{c^{2}}∇ & 0 & −Δ_{c^{2}}b(2^{n−1} − 1) & \frac{1+m}{2}cΔ_{c^{2}} & ρ \\ 0 & 0 & a^{2^{n−1}} & 0 & 0 & 0 & ∇ & 0 & (2^{n−1} − 1)b & (2^{n−1} − 1)(∇)^{2} \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \frac{1+m}{2}c∇^{m}Δ_{(c∇^{m})^{2}} \\ 0 & 0 & 0 & 0 & 0 & a^{2^{n−1}} & 0 & a^{2^{n−1}} & 0 & (2^{n−1} − 1)b∇^{2^{n}−1} \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & a^{2^{n−1}} & 0 & ∇ \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} )

Here, ρ is a solution of the following equation (this obtained from computing the (2,6)- position of the respective matrices)

−Δ_{c^{2}}b(2^{n−1})(c∇^{m} − 1) + \frac{1 + m}{2}cΔ_{c^{2}}(b∇^{2^{n}−1} − 1) + ρ(a − 1)

= (b − 1)\frac{1 + m}{2}cΔ_{c^{2}} + (1 + a^{2^{n−1}}c)Δ_{c^{2}}(2^{n−1} − 1)b

This is equivalent to

bcΔ_{c^{2}}((2^{n−1} − 1)(∇^{m} + a^{2^{n−1}} + \frac{1 + m}{2}(−∇^{2^{n}−1} + 1))) = ρ(1 − a)

Since the equation on the left hand side has augmentation 0 with respect to a, the equation is solvable in the augmentation ideal with respect to a, hence the existence of ρ is proved.

The corresponding equation for σ is (this is obtained from computing the (1,6)-position):

σ(a − 1) + \frac{1}{2}((c − 1)Δ_{c^{2}} + Δ_{cm})(c∇^{m} − 1) − λ∇^{2^{n−1}} = (ca^{2^{n−1}} − 1)\frac{1 + m}{2}cΔ_{c^{2}}

Since −\frac{1}{2}((c − 1)Δ_{c^{2}} + Δ_{cm})(c∇^{m} − 1) + λ∇^{2^{n−1}} + (ca^{2^{n−1}} − 1)\frac{1+m}{2}cΔ_{c^{2}} has augmentation 0 with respect to a, this equation is solvable for σ.

In d_{3} only λ might be of odd augmentation. The rest of the coefficients of Hom_{ℤG}(d_{3}, 𝔽_{2}) are 0 regarded as mapping between two copies of Hom_{ℤG}(Γ, 𝔽_{2}). Thus, if we write the elements of Hom_{ℤG}(Γ^{n}, 𝔽_{2}) ≅ (Hom_{ℤG}(Γ, 𝔽_{2}))^{n} as column vectors, then the image of Hom_{ℤG}(d_{3}, 𝔽_{2}) is contained in the 5-th row. The kernel of d_{4} has, for the same reason, a complement that is contained in the direct sum of the 8-th and the 9-th row. In any case, the 10 − th row (as well as the 6-th row) is fully contained in H^{3}(G, 𝔽_{2}). Because (2^{n−1}−1)b∇_{a}^{2^{n}−1} has odd augmentation, we get that M_{3} does not act trivially on H^{3}(G, 𝔽_{2}).

This finishes the proof of Theorem 1.

Remark 10 We remark that with the very same methods one can see that with G = C_{8} ×Q_{8} and Q_{8} being the quaternion group of order 8, and this acting on C_{8} via the natural epimorphism

Q_{8} ⟶ ⟶ C_{2} × C_{2} ≃ Aut(C_{8})

any non inner but almost inner automorphism of G induces a non trivial automorphism of H^{3}(G, 𝔽_{2}). This is interesting since in [9] one constructs a group

H = (C_{p}^{2} × C_{q}^{2}) ×(G ×C_{2})

and a non inner automorphism α of H which becomes inner in SH for either a suitably big ring of algebraic integers S or for any semilocalization S of ℤ. Of course, it follows that α is almost inner and thus α induces the identity on H^{∗}(H, ℤ).

We remark that D. Benson computes in [1] the first terms of a projective resolution of the trivial module ℤ for the group (C_{p} × C_{q}) ×Q_{8} in view of a possible counterexample to the Poincaré conjecture.

References

[1] D. Benson, Periodic resolutons for finite groups, preprint 1998.

[2] K. S. Brown, Cohomology of groups, Graduate Texts in Math. no 87 Springer Verlag, Berlin-New York, 1982.

[3] D. Coleman, On the modular group ring of a p-group, Proc. Amer. Math. Soc. 15 (1964) 511– 514.

[4] M. Hertweck, A counterexample to the isomorphism problem for integral group rings of finite groups, to appear in Comptes Rendus de l’Académie des Sciences de France.

[5] M. Hertweck, Eine Lösung des Isomorphieproblems für ganzzahlige Gruppenringe von endlichen Gruppen (1999) Shaker Verlag, ISBN 3-8265-6055-8

[6] B. Huppert, Endliche Gruppen, Grundlehren der mathematischen Wissenschaften Bd. 134, Springer Verlag Berlin-Heidelberg 1967.

Automorphisms on Cohomology 13

[7] S. Jackowski, problem 18 of the problem session in the ”American Mathematical Society summer research institute” Seattle 1996, Proceedings of Symposia in Pure Mathematics 63 (1998).

[8] S. Jackowski and Z. Marciniak, Group automorphisms inducing the identity on map on cohomology, J. Pure and Applied Algebra 44 (1987) 241-250.

[9] K. W. Roggenkamp and A. Zimmermann, Outer group automorphisms may become inner in the integral group ring, J. pure and appl. Algebra 103 (1995) 91-99.

[10] C. T. C. Wall, Resolutions for extensions of groups, Proc. Camb. Phil. Soc. 57 (1961) 251-255.

Address of the authors:

Eric Jespers

Department of Mathematics Vrije Universiteit Brussel Pleinlaan 2 1050 Brussel Belgium Alexander Zimmermann LAMFA Faculté de Mathématiques Université de Picardie Jules Verne 33 rue St Leu 80039 Amiens CEDEX France