Answer TWO questions

Deduce that if |G| = pⁿ and |H| = p^n-1 (p a prime), then H is a normal subgroup of G.

[Hint for second part: Let q be the action defined in the first part, let ker(q) be its kernel, and let q(G) be its range. From the First Isomorphism Theorem, deduce that |q(G)| = p^k, for some k. From q(G) £ S_X deduce that k £ 1. From the First Isomorphism Theorem again deduce that |ker(q)| ³ p^n-1.]

2. State the three theorems of Sylow and sketch the proof of one of them.

Let the group G have order 2pq, where p,q are odd primes with q > 2p and p ¹ 1 mod q. Prove that G has a normal subgroup of order q and a normal subgroup of order p.

How many elements of order q does G have? How many elements of order p does it have?

Give two examples of groups of order 2pq having different numbers of elements of order 2.

3. Let G be a group, Aut(G) its group of automorphisms, Inn(G) its group of innner automorphisms, and Z(G) its centre. Prove that

Now consider the dihedral group D₄, that is, the group

What is the centre of D₄? Deduce that Inn(D₄) @ D₄.

Write down the orders of all the elements of D₄.

Let q be an automorphism of D₄. Show that q(r) can only be r or r³ and that q(s) can only be one of rⁱs, 1 £ i £ 4.

Deduce that |Aut(D₄)| £ 8 and hence that Aut(D₄) @ Inn(D₄) @ D₄.