Thread: Abelian groups and automorphisms

Exercise 2 in Dummit and Foote Section 4.4 on Automorphisms reads as follows:

Prove that if G is an abelian group of order pq, where p and q are distinct primes, then G is cyclic.

I would appreciate help with this proof>

{I guess as the proof unfolds, the connection to automorphisms will become clear!}

Peter

you don't need automorphisms to prove this. use cauchy's theorem to produce an element of order p, and an element of order q, say a and b. consider what the order of ab could possibly be.

(this is, by the way, the heart of what is known as the chinese remainder theorem, since the only fact we really need is that gcd(p,q) = 1).