This is Q#9, Section 2.9, pg 75 from Herstien

Q: order(G) = pq, where p and q are distinct primes. G has a normal subgroup of order p and a normal subgroup of order q. Prove G is cyclic.

My attempt:

Let N1 be the subgroup of order p

Let N2 be the subgroup of order q

Clearly, N1 and N2 are cyclic (as for every group with prime order). Let a, b be generator of N1 and N2 respectively.

Consider c=ab

I can show that order(c)=pq. And, hence G is cyclic (with c as generator)

Help I need:

1. Is my attempt correct?

2. I never used the fact that N1 and N2 are normal (something mentioned in the question) - So am I missing / mis-understanding something?

Thanks