Group Theory: Sheet 4

\begin{matrix}c \\ ◯ \end{matrix}G. C. Smith 2001

The copyright is waived for non-profit making educational purposes. The course web site is available via http://www.bath.ac.uk/∼masgcs/

1. Let G be a group.

(a) Suppose that H ≤ G and |G : H| = 2. Prove that H ⊴ G.

(b) Show that S_{3} has a non-normal subgroup of index 3.

2. Let G be a group, and suppose that H, K are subgroups of G.

(a) Show that HK is a subgroup of G if and only if HK = KH.

(b) Suppose that K ⊴ G. Show that HK is a subgroup of G.

(c) Suppose that G is a finite group. Show that |HK|·|H ∩K| = |H|·|K|.

3. Suppose that G is a group and that N is a finite normal subgoup of G. Show that G must have a subgroup H of finite index with the property that every element of H commutes with every element of N. Hint: for each g ∈ G there σ_{g} ∈ Aut(N) defined by n ↦ g^{−1}ng for every n ∈ N.

4. Let G be a group.

(a) Show that x, y ∈ G commute if and only if x^{−1}y^{−1}xy = 1.

(b) Suppose that M, N are normal subgroups of G such that M ∩N = 1. Show that each element of M commutes with each element of N.

(c) Suppose that M, N are normal subgroups of G such that both G/M and G/N are abelian. Prove that G is an abelian group.

5. Show that Aut(S_{4}) is isomorphic to S_{4}.

6. (a) Suppose that G is a group. Show that the map i : G → G defined by g ↦ g^{−1} for every g ∈ G is an automorphism of G if and only if G is abelian.

(b) Let G be a group with the property that g^{2} = 1 for every g ∈ G. Prove that G must be abelian, and show that G (written additively) may be regarded as a vector space over the field 𝔽_{2} with two elements.

(c) Suppose that G is a finite group with a trivial automorphism group. Prove that G has either 1 or 2 elements.