Group Theory: Sheet 5

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

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1. Let G be a group, and let S be a subset of G. Let

C_{G}(S) = {g ∈ G | g^{−1}sg = s∀s ∈ G} and N_{G}(S) = {g ∈ G | g^{−1}Sg = S}.

(a) Prove that C_{G}(S) ≤ N_{G}(S) ≤ G.

(b) Prove that C_{G}(S) ⊴ N_{G}(S).

(c) Show that N_{G}(S)/C_{G}(S) is isomorphic to a subgroup of SymS.

2. Let G be a finitely generated group in which for each x ∈ G, the set S_{x} = {g^{−1}xg | g ∈ G} is finite. Let Z(G) denote the centre of G (the subgroup of G consisting of those elements of G which commute with all elements of G). Prove that Z(G) has finite index in G. (Hint: Do Question 1 first)

3. Prove that Aut S_{5} is isomorphic to S_{5}.

4. Suppose that G = ⟨x, y⟩ and x^{2} = y^{2} = 1. Prove that G has a cyclic subgroup of index at most 2.

5. Suppose that G = ⟨S⟩ is a group and that S is finite. Now suppose that also G = ⟨T ⟩. Show that there is a finite subset U of T such that G = ⟨U⟩.

6. Consider the maps α, β ∈ Sym ℤ defines by α : i ↦ i + 2 for all i ∈ ℤ, and β = (0, 1). Thus β fixes all integers except 0 and 1, and it swaps those. Consider the groups G = ⟨α, β⟩ and H = ⟨T ⟩ where T = {α^{−k}βα^{k} | k ∈ ℤ}. Observe that all elements of H have finite support (i.e. each one moves only finitely many integers). Deduce that a finitely generated group may have a subgroup which has no finite generating set.

7. Let G be a finite cyclic group of order n. Suppose that d is a divisor of n. Show that G has a unique subgroup of order d.