Firstly observe that for a general group,

So I think your identity should read , unless the problem mentioned that the group is commutative.

Have they asked you to use Induction? You can do without it...

You can do likewise for a.

Let the order of (ab) be x. Now, since . But p and q are primes, the only divisors of pq are 1,p,q and pq. Its easy to see that if x=p, then ...Contradiction.

Similarly for x=q. Thus we are left with only pq.