Let G be a group of order pq where p and q are primes and p

Let G be a group of order pq where p and q are primes and p > q. Prove that: G has a normal subgroup of order p.

Solution

As per Lagrange theorem, if H is a subgroup of G, the order of H divides the order of G.

If we assume that H is non-trivial (not equal to {e} or G), then we have that |H| is a divisor of |G| i.e. |H| is a divisor of pq. This implies that either |H| is either a divisor of p or a divisor of q as both p, q are primes. Further, since H is assumed to be non-trivial, this means that either |H|=p| or |H|=q. Thus, G has a subgroup of order p and a subgroup of order q. Now, by the Sylow theorems, the subgroups of order p and q are both normal in G. Thus, G has a normal subgroup of order p.