Group of order pq must have subgroups of order p and q.

Hi:

Notation:

o(G) order of group G.

(a) subgroup generated by a.

I have a problem which is stated as follows: let o(G) be pq, where p,q are primes, p > q. Then G has a subgroup of order p and a subgroup of order q.

I have been able to prove G has a subgroup of order q, say (b). Then I have taken a not belonging to (b) and I have seen (a) intersection (b) = (e). What I say is this: if o(a) = p the statement is proved. If o(a) = q,

I say o(ab) = p. But I cannot prove it. Any hint would be welcome. Thanks for reading.