Prove that no group of order pq, where p & q are primes, is simple.?

Answer 1

Let G be such a group. We assume (without loss of generality) that p<=q. Then consider the Sylow q-subgroups of G. The number of such groups must divide pq, and so must be 1, p, q, or pq. This number must also be 1 (mod q), which leaves us with 1 as our only choice. Thus there is a unique Sylow q-subgroup, and we know that a unique Sylow subgroup is normal, so we have that G is not simple.