I have a question concerning a group of order pq, where p,q are distinct prime numbers.
Does such a group necessarily have a normal subgroup of order p, and a normal subgroup of order q?
Thank you !
Yes, this follows from Sylow's Third Theorem. If n is the number of Sylow p-subgroups in G, then n = 1 (mod p) and n|pq. However, the only value for n that satisfies both of these conditions is 1. So, there is only one Sylow p-subgroup in G, meaning it is normal. The same is true for the single Sylow q-subgroup.
This isn't quite accurate: there will always be a normal Sylow subgroup of order the greatest prime, but the Sylow sbgp. of order the lowest prime may not be normal. For example, $\displaystyle S_3$ , of order $\displaystyle 6=2\cdot 3$ , has a normal subgroup of order 3 but no normal sbgp. of order 2.
Tonio