GROUP COHOMOLOGY

1) If G is a profinite group and H is a closed subgroup show that Ind _{H}^{G} is an exact functor from Mod_{H} to Mod_{G}.

2) If G = {±1} find an injective G-module which contains ℤ with G acting trivially. Also find an injective G-module which contains ℤ/2ℤ with trivial G-action.

3) If G is a profinite group and H is an open subgroup show that cor_{H}^{G}res_{H}^{G} = [G : H].

4) Let G be a profinite group and H a closed subgroup.

(a) Show that under the canonical isomorphism H^{i}(G, Ind _{H}^{G}M) ≅ H^{i}(H, M) the restriction map H^{i}(G, M) → H^{i}(H, M) becomes identified with the map H^{i}(G, M) → H^{i}(G, Ind _{H}^{G}M) induced by

\begin{matrix}M & ⟶ & Ind _{H}^{G}M \\ m & ⟼ & (g ↦ gm). \end{matrix}

(b) If H is open in G, show further that the corestriction map H^{i}(H, M) → H^{i}(G, M) becomes identified with the map H^{i}(G, Ind _{H}^{G}M) → H^{i}(G, M) induced by

\begin{matrix}Ind _{H}^{G}M & ⟶ & M \\ φ & ⟼ & ∑_{g∈H\G} g^{−1}φ(g). \end{matrix}

5) Let G be a profinite group, H a closed subgroup and

(0) ⟶ M ⟶ N ⟶ P ⟶ (0)

an exact sequence of G-modules. Let ϕ ∈ Z^{i}(G, P).

(a) Find an element of Z^{i}(H, P) representing res_{H}^{G}[ϕ] ∈ H^{i}(H, P).

(b) Find an element of Z^{i+1}(G, M) representing δ_{i}[ϕ] ∈ H^{i+1}(G, M).

6) Let G = ℤ̂ = lim_{←N} ℤ/Nℤ and let F ∈ G correspond to 1 ∈ Ẑ. Let M denote a torsion G-module.

(a) Show that

\begin{matrix}Z^{1}(G, M) & ⟶ & M \\ φ & ⟼ & φ(F) \end{matrix}

is an isomorphism. Hence show that H^{1}(G, M) ≅ M/(F − 1)M.

(b) Show that for i > 1 we have H^{i}(G, M) = (0). [Hint: Use dimension shifting.]

7) Give an example of a finite group G and a short exact sequence of G-modules

(0) ⟶ M ⟶ N ⟶ P ⟶ (0)

such that

(0) ⟶ M^{G} ⟶ N^{G} ⟶ P^{G}

is not right exact.

8) Let G be a profinite group and M and N discrete G-modules. By an extension of N by M which is split as an abelian group we mean a short exact sequence of G-modules

(0) ⟶ M ⟶ E ⟶ N ⟶ (0)

such that there is a homomorphism of abelian groups N → E which is a section to E → → N. We call two such extensions E and E′ equivalent if there is an isomorphism E →^{∼} E′ compatible with the injections M ↪ E and M ↪ E′ and with the surjections E → → N and E′ → → N. Make Hom (N, M) into a G-module by setting g(f) = g ∘ f ∘ g^{−1}. Show that there is a bijection bewteen equivalence classes of extensions of N by M which split as abelian groups and H^{1}(G, Hom (N, M)). [Hint: If ϕ ∈ Z^{1}(G, Hom (N, M)) define a G-module structure on M ⊕ N by setting

g(m, n) = (gm + ϕ(g)(n), gn).]

If N is a trivial G-module, show that the element in H^{1}(G, Hom (N, M)) parametrising E is the image of δ_{0}(E) ∈ Hom (N, H^{1}(G, M)) under a natural map

Hom (N, H^{1}(G, M)) ⟶ H^{1}(G, Hom (N, M)),

which you should describe.

9) Let M and G be finite groups. By an extension of G by M we mean an exact sequence of groups

1 ⟶ M ⟶ E ⟶ G ⟶ 1

(where M embeds as a normal subgroup of E). We call two such extensions E and E′ equivalent if there is an isomorphism E →^{∼} E′ compatible with the injections M ↪ E and M ↪ E′ and with the surjections E → → G and E′ → → G. Write Aut (M) for the group of automorphisms of M. There is a natural homomorphism c : M → Aut (M) defined by

c(m)(n) = mnm^{−1}.

The image of c is a normal subgroup Inn(M) of Aut (M) and we write Out (M) for the quotient Aut (M)/Inn(M).

(a) Show that there is a natural bijection between equivalence classes of extensions of G by M and equivalence classes of pairs (ϕ, ψ), where ϕ : G → Aut (M) and ψ : G × G → M satisfy

ϕ(g)ϕ(h)ϕ(gh)^{−1} = c(ψ(g, h))

and

ψ(g, h)ψ(gh, k) = ϕ(g)(ψ(h, k))ψ(g, hk)

for all g, h, k ∈ G. Here we consider (ϕ, ψ) and (ϕ′, ψ′) equivalent if there is a map α : G → M such that

ϕ′(g) = c(α(g))ϕ(g)

and

ψ′(g, h) = α(g)ϕ(g)(α(h))ψ(g, h)α(gh)^{−1}.

[Hint: Define a group structure on the set M × G by

(m, g)(n, h) = (mϕ(g)(n)ψ(g, h), gh).]

(b) If H is any finite group and f : H → E is a homomorphism, show that all homomorphisms H → E with the same projection to G as f are parametrised by H^{1}(H, M), where H acts on M via

H ⟶^{f} E ⟶ Aut (M).

(c) If M is abelian show that there is a natural bijection between equivalence classes of extensions of G by M and pairs (ϕ, [ψ]), where ϕ ∈ Hom (G, Aut (M)) and [ψ] ∈ H^{2}(G, M) with G acting on M via ϕ.

(d) If M is abelian and \overline{f} : H → G is a homomorphism, show that \overline{f} lifts to a homomorphism f : H → E if and only if the pull back (via \overline{f} - restriction followed by inflation) of [ψ] to H^{2}(H, M) is zero.

(e) If M has trivial centre show that there is a natural bijection between equivalence classes of extensions of G by M and Hom (G, Out (M)). Show also that the extension corresponding to ϕ ∈ Hom (G, Out (M)) is

{(α, g) ∈ Aut (M) × G : αInn(M) = ϕ(g)}.

(f) If E is a finite group and M is a minimal normal subgroup show that either M is abelian or M has trivial centre. [In fact M ≅ S^{r} for some simple group S and some r ∈ ℤ_{>0}. You can try to prove this too if you like. Thus if you understand finite simple groups and a little group cohomology you can in some sense reconstruct all finite groups.]

10) Let p be a rational prime.

(a) Show that there is a matrix A ∈ M_{2}(Z_{p}) with

(( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) + pA)^{p} ≡ 1_{2} mod p^{2}

if and only if p = 2 or 3. Show further that if p = 2 or 3 then we can choose A so that

(( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) + pA)^{p} = 1_{2}.

(b) Show that the exact sequence

1 ⟶ M_{2}(ℤ/pℤ) ⟶ GL_{2}(ℤ/p^{2}ℤ) ⟶ GL_{2}(ℤ/pℤ) ⟶ 1

splits (i.e. admits a section GL_{2}(ℤ/pℤ) → GL_{2}(ℤ/p^{2}ℤ) which is a homomorphism) if and only if p = 2 or 3. Show further that if p = 2 or 3 then

1 ⟶ 1_{2} + pM_{2}(ℤ_{p}) ⟶ GL_{2}(ℤ_{p}) ⟶ GL_{2}(ℤ/pℤ) ⟶ {1}

also splits. [Hint: ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) generates a Sylow p-subgroup of GL_{2}(ℤ/pℤ).]