Group Theory: Sheet 6

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

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1. Let G be a group and let G act on itself by conjugation. Choose a transversal {x_{i} | i ∈ I} for the conjugacy classes.

(a) Suppose that G is finite. Prove that

|G| = ∑_{i∈I} |G : C_{G}(x_{i})|

by applying the orbit-stabilizer theorem.

(b) Suppose that G is finite of order p^{n} where n ≥ 1, Use part (a) to show that |Z(G)| ≡ 0 mod p and deduce that |Z(G)| > 1.

2. Let G be a group of order p^{2} where p is a prime number.

(a) Show that if G is non-abelian, we must have |Z(G)| = p and g^{p} = 1 for every g ∈ G.

(b) Suppose that G is non-abelian, and choose h ∈ G with h ∉ Z(G). Let H = ⟨h⟩. Show that H ≤ N_{G}(H) and Z(G) ≤ N_{G}(H), and deduce that H ⊴ G.

(c) Prove that Z(G)H = G and Z(G)∩H = 1. Deduce that every group of order p^{2} is abelian.

3. Let G be a finite group of even order. Show that the number of elements which do not have order 2 is even, and deduce that G contains an element of order 2.

4. Let G be a finite group and suppose that p is a prime number which divides |G|. Let

Ω = {(g_{1}, g_{2}, …, g_{p}) | g_{1}g_{2} … g_{p} = 1}.

(a) Prove that |Ω| = |G|^{p−1}.

(b) Let H = ⟨x⟩ be a cyclic group of order p. Define a multiplication by (g_{1}, g_{2}, …, g_{p}) · x^{i} = (g_{1+i}, g_{2+i}, …, g_{p+i}) for each (g_{1}, g_{2}, …, g_{p}) ∈ Ω. Show that every product (g_{1}, g_{2}, …, g_{p})·x^{i} is in Ω and demonstrate that this multiplication gives an action of H on Ω.

(c) Show that each orbit of H acting on Ω has size 1 or p.

(d) Describe the orbits of H on Ω which have size 1.

(f) Use the orbit-stabilizer theorem to prove Cauchy’s result that if a prime number p divides the order of a finite group, then there is g ∈ G with g of order p (i.e. |⟨g⟩| = p).

(g) How does Question 4(f) relate to Question 3?

5. Suppose that H ≤ G. Show that the number of conjugates subgroups of the form H^{g} (as g ranges over G) is a divisor of |G|.

6. Suppose that G is a group and that H is a subgroup of G. If g ∈ N_{G}(H), show that τ_{g} : H → H defined by h ↦ h^{g} for every h ∈ H is an automorphism of H. Show that τ : g ↦ τ_{g} is a homomorphism from N_{G}(H) to Aut (H). Deduce that C_{G}(H) ⊴ N_{G}(H) and that N_{G}(H)/C_{G}(H) is isomorphic to a subgroup of Aut (H).