I have done the classification of groups of order $p^2q$, where $p>q$. $p,q$ odd distinct prime
If $P\in Syl_{p}(G)$ & $Q\in Syl_{q}(G)$ then Case $1$: $P=\Bbb Z_{p^2}$. I have done.
But my question is for Case $2$: $P=\Bbb Z_{p}\times \Bbb Z_{p}$
Here $\phi: Q \to Aut(P)$
s.t $q\mapsto \phi(q)$ where $\phi$ is injective. Then as q is odd prime $o(\phi(q))|(p-1)p(p+1)$ [Because we can retrict $\phi$ to $SL_{2}\Bbb (Z_p)$
Now $q \not|p$ So $q | (p-1)(p+1) \Rightarrow q|(p+1)$ or $q|(p-1)$
Now Can anyone help me from here. I am also trying.
Actually 1st problem is finding a Subgroup H $\in$ $SL_{2}\Bbb (Z_p)$ of order $(p+1)$ & $(p-1)$. Then cases for $q$.
- One more thing before giving this problem. I was observing some related question on this in mathstack. But I couldn't understand many notation & result. Perhaps out of my knowledge. So please don't refer any other answer & give the answer on the basis of Semidirect Product & explain it thoroughly. Thanks in advance.
Hint
Consider the field $F$ with $p^{2}$ elements, and an element $z \in F$ of order $p^{2}-1$. Now $z$ acts linearly by multiplication on the additive group of $F$, which is isomorphic to $P$. This will yield an automorphism of $P$ of order $p^{2} - 1$.