1. group of order pq

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 !

2. 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.

3. Originally Posted by spoon737
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, $S_3$ , of order $6=2\cdot 3$ , has a normal subgroup of order 3 but no normal sbgp. of order 2.

Tonio

4. Originally Posted by tonio
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, $S_3$ , of order $6=2\cdot 3$ , has a normal subgroup of order 3 but no normal sbgp. of order 2.

Tonio
Good point, I was a little too hasty in my answer.

5. Originally Posted by spoon737
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.
Not correct. Even if p,q are both odd, e.g. p=5,q=11, you could have a group with 11 Sylow 5-subgroups. However it's true that there's only one Sylow q-subgroup if q>p

6. thank you.
It will take some time for me to understand what you said, but thank you