Math 551, Assignment 4

Due Thursday, Oct. 12 in class

1. Let K ◁ H ≤ G and N ◁ G. Show that KN ◁ HN and that HN/KN is a quotient of H/K (that is, there exists a surjective homomorphism H/K → HN/KN).

2. Let N be the set of natural numbers. Let Σ_{N}^{o} be the set of all permutations of N with finite support. Define A_{N}^{o} and show that A_{N}^{o} is simple. (Hint: represent A_{N}^{o} as a union of finite alternating groups.)

3. Let N ◁ G. Show that G has a composition series if and only if both N and G/N have composition series.

On the other hand, give an example of a group G and a subgroup H ≤ G such that G has a composition series but H does not.

4. If K ◁ G and L ◁ G, show that G/(K ∩ L) is isomorphic to a subgroup of G/K × G/L.

5. Suppose H_{1}, …, H_{n} are subgroups of G and |G : H_{i}| is finite for each i = 1, …, n. Show that the index

|G : ⋂_{i=1}^{n} H_{i}|

is finite and give a bound for it which is a function of the indices |G : H_{1}|, …, |G : H_{n}|.

6. Let G be any simple group such that |G| = 60. Let H = A_{6}.

a) Construct an injective homomorphism ϕ : G → H such that for any element x ∈ G of order 3, ϕ(x) is the product of two disjoint 3-cycles.

b) Then construct an injective homomorphism ψ : H → A_{H/ϕ(G)} such that every element of ψ(ϕ(G)) fixes at least one point (of H/ϕ(G)).

c) Conclude that any simple group of order 60 is isomorphic to A_{5}.

d) Conclude also that A_{6} has an automorphism α such that α((123)) = (123)(456). (Thus α is not induced by conjugation by any element of Σ_{6}. !)

7. Show that A_{5} contains elements x, y and z of orders 2, 3 and 5 respectively such that xyz = 1. On the other hand, show that no solvable group can contain such a triple of elements. Generalize the second statement from (2, 3, 5) to other triples of integers.

8. Given two groups K and H and a homomorphism ϕ : H → Aut(K), one can define a group G = H ×_{ϕ} K (the semidirect product of K by H via ϕ) as follows: the underlying set of G is the Cartesian product K × H, and the multiplication is

(k_{1}, h_{1})(k, h) = (k_{1}ϕ(h_{1})(k), h_{1}h).

(The idea is that G should consist of “products” kh, k ∈ K, h ∈ H, with the rule hkh^{−1} = ϕ(h)(k) for moving h’s past k’s.)

Verify that G is a group, and there are two subgroups of G which can be identified with K and H in such a way G = KH, K ∩ H = 1 and K ◁ G; moreover ^{h}k = ϕ(h)(k) for all h ∈ H and k ∈ K. Usually the notation H × K is used, even though it omits essential information.

The holomorph of a group G is defined to be the semidirect product Aut(G) × G, with ϕ being the the identity mapping. What is the holomorph of Z_{2} × Z_{2}?