COHOMOLOGY OF GROUPS : EXERCISES

Sheet 3, 26 April 2007

Exercise 3.1. Compute H^{2}(ℤ/2, ℤ/4) for any action of ℤ/2 on ℤ/4. Is the result compatible with your solution of Exercise 2.7 ?

Exercise 3.2. Let (G, N, ϕ) be an abstract kernel, and C be the center of N with G-action induced by ϕ. Prove that the group H^{2}(G, C) acts freely and transitively on the set of isomorphism classes of extentions E(G, N, ϕ) (see the hints given in the lecture).

Exercise 3.3. Let G be a group different then ℤ/2, and C be a G-module. Prove that any element α of H^{3}(G, C) can be represented as the obstruction to realizing an abstract kernel (G, N, ϕ).

Hint: Let F be the (non-commutative) free group generated by the set

{ [x, y] | x, y ∈ G − {1} } ,

and take N = C × F (it has center C). Define f : G × G → N = C × F by f(x, y) = (1, 1) if x = 1 or y = 1, and f(x, y) = (1, [x, y]) otherwise. Let k : G^{3} → C be a 3-cocycle in hom_{ℤG}(F̄_{G}, C) representing α. Define a function L : G → Aut(N) by

L(x)(c, [y, z]) = (c + k(x, y, z), f(x, y)f(xy, z)f(x, zy)^{−1}),

and show that it satisfies L(x)L(y) = τ(f(x, y))L(xy). In particular L induces a homomorphism ϕ : G → Out(N). Show that the obstruction to realizing (G, N, ϕ) is α. (If you are interested in a proof for the case G = ℤ/2, see “S. Eilenberg and S. Mac Lane, Cohomology theory in abstract groups. II. Group extensions with a non-Abelian kernel. Ann. of Math. (2) 48, 1947 326–341”).

Exercise 3.4. We saw in the lecture that H^{2}(G, C) is a functor in two variables. How is this functoriality transposed in the (non-canonical) interpretation of H^{2}(G, C) as extentions ?

Exercise 3.5. Classify all extentions 1 → ℤ/4 → E → ℤ/4 → 1 up to isomorphism, and give an explicit representative for each isomorphism class.

Exercise 3.6. Classify all extentions 1 → Q_{8} → E → ℤ/2 → 1 up to isomorphisms, and give an explicit representative for each isomorphism class.

Hint: Recall that Q_{8} admits the presentation ⟨a, b|a^{4} = 1, b^{2} = a^{2}, bab^{−1} = a^{3}⟩, and show that Aut(Q_{8}) ≅ S_{4} and Out(Q_{8}) ≅ S_{3}, where S_{n} denotes the group of permutations of a set with n elements.

Typeset by AMS-TEX