I am struggling with semidirect products and how they can be used to classify groups of a certain order. In particular, I need help with the nonabelian case. This is the problem I am working with..
Classify all groups of order $pq^2$ with $p$,$q$ primes, $p<q$, $p\nmid(q-1)$, and $p^2\nmid(q+1)$. Use can use the fact that $GL_2(\mathbb{Z}_q)$ has $(q^2-1)(q^2-q)$ elements.
Ok. So here's my thought process. I first considered when $G$ was abelian and applied the Fundamental Theorem of Finitely Generated Abelian Groups (FTFGAG) to obtain all abelian groups of this order. My results were: $\mathbb{Z}_{pq^2}$ and $\mathbb{Z}_{pq}\times\mathbb{Z}_q$.
Next considered when $G$ was nonabelian and applied Sylow's Theorem to determine how many Sylow $p$ and $q$ subgroups there were in $G$. I found the Sylow $q$-subgroups to be unique, and hence normal. I let the Sylow $q$-subgroup be called $H$. I let $K$ be any Sylow $p$-subgroup in $G$. Then by Lagrange, $H\cap K=1$. Next I showed that $G=HK$ and let $\varphi: K\rightarrow \text{Aut}(H)$ be a homomorphism. Then by applying theorem 12 from Dummit and Foote (sorry it didn't have a name :/), I got $G\cong H\rtimes_\varphi K$.
Now I just need to consider all isomorphisms of $H$. They are $\mathbb{Z_{q^2}}$ and $\mathbb{Z}_q\times\mathbb{Z}_q$.
Suppose $H=\mathbb{Z_{q^2}}$. Then $|\text{Aut}(H)|=q(q-1)$. Since $p\nmid q$ and $p\nmid q-1$, there does not exist an element of order $p$ in $\text{Aut}(H)$ by Lagrange. This means the only homomorphism is trivial. Therefore $H\rtimes_\varphi K\cong \mathbb{Z}_{q^2}\times \mathbb{Z}_p$. But this is abelian and contradicts my assumption that $G$ is nonabelian. Plus FTFGAG, already classified all abelian groups. Therefore $H=\mathbb{Z}_{q^2}$ does not result in a new group.
And here is where I start to get lost...
Suppose $H=\mathbb{Z_q}\times\mathbb{Z_q}$. Then $|\text{Aut}(H)|=(q^2-1)(q^2-q)$. I do not see how $p$ divides this...
Help with this last part would be much appreciated. Thanks.
Also here are some resources I have looked at: https://crazyproject.wordpress.com/2010/06/25/classify-the-groups-of-order-75/