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## Homework Statement

Let p and q both be prime numbers and p > q. Classify groups of order p

^{2}q if p is not congruent to +1 or -1 mod q.

## Homework Equations

## 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 p

^{2}& congruent to 1 mod 1. And the number of subgroups of order p

^{2}must be 1 or q and congruent to 1 mod p.

I think there has to be 1 normal subgroup of order p

^{2}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 p

^{2}q.