Group Theory: Sheet 3

\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. Do exercises 1.11, 1.12, 2.1 to 2.7 in Topics in Group Theory. Only look at the answers at the back of the book if you get really stuck.

2. Suppose that α : G → H is an isomorphism of groups. Show that the map α^{−1} : H → G is also an isomorphism.

3. (a) Suppose that α is an automorphism of a group G. Let F_{α} = {g ∈ G | (g)α = g}. Prove that F_{α} ≤ G.

(b) Let G = GL(n, ℂ) be the group of invertible n by n complex matrices. Consider the map β : G → G which sends each matrix to the transpose of its inverse. Prove that β is an automorphism of G. Deduce that the subset of G consisting of those g ∈ G such that gg^{T} = I_{n} is a group.

4. Determine all automorphisms of the additive group ℤ. Determine all homomorphisms γ : ℤ → ℤ.

5. Determine all automorphisms of the additive group ℚ of rational numbers.

6. List all the automorphisms of S_{3}, and demonstrate that you have found them all.

7. Suppose that H is a subgroup of a group G. Let Ω = G/H so Ω is a collection of subsets of G. Subsets of G may be multiplied together. Suppose that Ω is closed under multiplication. Prove that Ω is a group, and deduce that H is a normal subgroup of G.

8. Suppose that G is a group with subgroups H and K. Suppose that both H and K have finite index in G. Deduce that H ∩ K must have finite index in G.

9. Exhibit a group G with the following property: the intersection of all subgroups of finite index in G is a subgroup of infinite index in G.