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.
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.
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?