Group Theory: Math30038, Sheet 4

GCS

1. Suppose that G acts on a set Ω. If α ∈ Ω, we let

G_{α} = {g ∈ G | αg = α}.

Now suppose that β, γ ∈ Ω are such that βh = γ for some h ∈ G. Show that G_{γ} = h^{−1}G_{β}h.

2. Let P be a group of order p^{n} where p is a prime number. Suppose that P acts on a finite set Q of size q where p does not divide q. Show that this action of P has a fixed point (i.e. there is α ∈ Q such that αg = α∀g ∈ P).

3. In how many essentially different ways can one colour the edges of a regular octahedron using c colours (where each edge is monochromatic, and two colourings are deemed the same if one can moved to the other by a rigid motion – and reflections are not allowed).

4. Let G be a group with subgroups H and K, each of finite index in G. Prove that H ∩ K has finite index in G.

5. Let G be a group and suppose that H ≤ G and |G : H| < ∞. By considering the groups g^{−1}Hg as g ranges over G (or otherwise), prove that G has a normal subgroup N with |G : N| < ∞ and N ≤ H ≤ G.

6. Let G be a group and suppose that x, y ∈ G. Prove that o(xy) = o(yx).