COHOMOLOGY OF GROUPS : EXERCISES

Sheet 2, 16 April 2007

Exercise 2.1. Let G be a group. Exhibit a contracting homotopy for the augmented bar resolution F_{G} → ℤ and the augmented normalized bar resolution F̄_{G} → ℤ, in the category of chain complexes of ℤ-modules. Do such homotopies exist in the category of chain complexes of ℤ[G]-modules ?

Exercise 2.2. Let g_{1}, …, g_{n} be elements in a group G that pairwise commute. Let S_{n} be the group of permutations of a set with n elements. Show that

z(g_{1}, …, g_{n}) = ∑_{σ∈S_{n}} sign(σ)[g_{σ(1)}| … |g_{σ(n)}]

is a cycle in ℤ ⊗_{ℤ[G]} F_{G}.

Exercise 2.3. Let p ⩾ 3 be a prime, C_{p} be the cyclic group of order p generated by an element t, and ℤ be given the trivial C_{p}-module structure (i.e., t is the identity of ℤ). Show that

H_{n}(C_{p}; ℤ) ≅ \{ \begin{matrix}ℤ & if n = 0, \\ ℤ/p & if n ⩾ 1 is odd, \\ 0 & otherwise. \end{matrix}

For n ⩾ 1 odd, find a representant of a generator of H_{n}(C_{p}; ℤ) in ℤ ⊗_{ℤ[G]} F_{G}.

Exercise 2.4. Let G be a group and M a ℤ[G]-module with G-action given by a homomorphism ϕ : G → Aut(M). Let C be the set of subgroups H of M ⋊_{ϕ} G such that H ∩ M = {1} and MH = M ⋊_{ϕ} G. We define an equivalence relation ∼ on C by setting H ∼ H′ if H and H′ are conjugate subgroups of M ⋊_{ϕ} G. Construct an isomorphism of sets

H^{1}(G, M) ⟶^{≅} C/ ∼ .

Hint : represent elements of H^{1}(G, M) by cocycles f : G → M in the cocomplex hom_{ℤ[G]}(F_{G}, M)

Exercise 2.5. Describe Aut(ℤ/n) for n ⩾ 2.

Exercise 2.6. Let n ⩾ 3 be an integer and let D_{2n} be the group of symmetries of a regular polygon with n sides in ℝ^{2}. It is called the dihedral group of order 2n. Show that there is an isomorphism

D_{2n} ≅ ℤ/n ⋊_{ϕ} ℤ/2

for ϕ : ℤ/2 → Aut(ℤ/n) sending the generator to multiplication by −1.

Exercise 2.7. Classify all extensions 1 → ℤ/4 → E → ℤ/2 → 1 up to isomorphism.

Typeset by AMS-TEX