Applying Sylow theorem for finding Normal Subgroup
Sylow Theorem: Let |G|=p^{\alpha}m, for some prime p such that p^{\alpha +1} \nmid |G|. If n be the number of p- Sylow subgroup(subgroup of order p^{\alpha}), then it will satisfy the following condition
n \mid m, and
n = 1 ~(\!\!\!\mod p).
Any subgroup of unique order is normal.
Classification of group of order pq: If G is a group of order pq, for primes p<q then we can have following possibilities
If p \nmid (q-1), then only cyclic group of order pq is possible.
If p \mid (q-1), then we get one cyclic group of order pq and one non-abelian group of order pq
G is a finite group of order p^2q where in p and q are distinct primes such that q \nmid (p^2-1) and p \nmid (q-1). Then G is an abelian group.