Let $p,q \in \mathbb{N}$ be prime numbers with the properties
$2 < p < q$ and $q - 1 , q + 1 \notin \left\langle p \right\rangle$
Classify all groups of the order $p^2q^2$ up to isomorphism.
This was a question given by my algebra professor and quite frankly I am stumped. My initial thought was that this question is referring to p-Sylow subgroups and one would need to apply the Sylow-theorems. If this is true, how would you apply them? Then what does "up to isomorphism" exactly mean?
I also thought to try and break it down and look at different possible cases. For example something like this:
Since $$q - 1 \notin \left\langle p \right\rangle \Rightarrow p \nmid q - 1 $$ $\Rightarrow \exists! $ subgroup of order $p$ $\Rightarrow \exists p - 1 $elements of order $p$ and $q-1$ elements of order $q$.
But honestly I am not sure how to answer this question. I would really appreciate if someone could try and explain this to me. Thank you in advance!