Group Theory: Math0038, Sheet 1

GCS

The course web site is available via http://www.bath.ac.uk/∼masgcs/ 1. Let G be a group. Suppose that x ∈ G. Let

C_{G}(x) = {g ∈ G | gx = xg} ⊆ G.

Prove that C_{G}(x) ≤ G.

2. Let G be a group. Suppose that S ⊆ G. Let

C_{G}(S) = {g ∈ G | gs = sg∀s ∈ S}.

Prove that C_{G}(S) ≤ G.

3. Let G be a group. Suppose that S ⊆ G. Let

N_{G}(S) = {g ∈ G | gS = Sg}

where gS = {gs | s ∈ S} and Sg = {sg | s ∈ S}. Prove that C_{G}(S) ≤ N_{G}(S) ≤ G.

4. Let G be a group. Suppose that S ⊆ G and that x ∈ N_{G}(S). Prove that C_{G}(S)x = xC_{G}(S).

5. Let G be a finite group. Suppose that ∅ ≠ H ⊆ G has the property that if a, b ∈ H, then ab ∈ H. Does it follow that H ≤ G? What happens if we relax the condition that G is finite?

6. Suppose that G is a group and that x ∈ G. Prove that (x^{−1})^{−1} = x.

7. Does there exist a group G containing elements a, b such that a^{2} = b^{2} = (ab)^{3} = 1?

8. Suppose that G is a group with the property that x^{2} = 1 whenever x is an element of G. Show that G must be abelian.

9. (Challenge) Suppose that G is a group with the property that x^{3} = 1 whenever x is an element of G. Show that G need not be abelian.