If a group has order p2q where p 1 and q 1 are primes prov

If a group has order p^2q where p > 1 and q > 1 are primes, prove that G must contain a proper normal subgroup.

Solution

et Nq be the number of Sylow q-subgroups. Then Nq=1or p or p2.

Suppose Nq=p. Since p1(mod q), p>q. Similarly if the number of Sylow p-subgroup is q, q>pq>p. This is a contradiction. Hence there is only one Sylow p-subgroup.

Suppose Nq=p2. Then the number of elements of order q is p2(q1). Hence the number of elements of order not equal to q is p2qp2(q1)=p2. Hence there is only one Sylow p-subgroup.