# Classifying groups using Sylow theorems

## Homework Statement

Let p and q both be prime numbers and p > q. Classify groups of order p2q if p is not congruent to +1 or -1 mod q.

## The Attempt at a Solution

It is clear that the Sylow theorems would be the things to use here. So I guess this says that the number of subgroups of order q must be either 1, p, or p2 & congruent to 1 mod 1. And the number of subgroups of order p2 must be 1 or q and congruent to 1 mod p.
I think there has to be 1 normal subgroup of order p2 since only one satisfies those conditions.
For subgroups or order q it seems that there can be only one normal subgroup as well.

What I'm not sure about is how to go about find which groups are isomorphic to these groups of order p2q.

VKint
If H and K are normal in the group G, |H| |K| = |G|, and H intersect K is trivial, what is the structure of G?

Well, I think G has to be abelian. So my hunch is that G will be isomorphic to Zp2 x Zq or Zp x Zp x Zq.

VKint
Correct. What I was getting at before was that, in general, if H and K are normal in G, |H| |K| = |G|, and H intersects K trivially, then G ~ H x K (where ~ denotes isomorphism and x is a direct product). This is easy to prove. First show that HK = G, and then construct the most obvious homomorphism possible between H x K and G. The other conditions on H and K will show that this homomorphism is well-defined and injective. You should then be able to verify that the conditions of this lemma are met by Hp2 and Hq in G (where H denotes a Sylow subgroup).

However, you're not done yet. There's only one isomorphism type for groups of order q (i.e., cyclic). How may types are there for groups of order p2?

Edit: I think I misunderstood your notation. Does "Zr" mean the cyclic group of order r?