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First let me give a general form of my question, and then I'll give some motivation and a more specific version of it. Let $K/k$ be a Galois extension of fields with Galois group $G$, and let $X$ be ...

Let $p,q$ be primes s.t. $q=np+1$. Denote $m=p^2$. Then $\mathbb{Z}_p$ acts non-trivially on $\mathbb{Z}_q$, so we have a non-abelian semi-direct product $\mathbb{Z}_q\rtimes \mathbb{Z}_m$, with the ...

Let $K$ be a field of characteristic $p$. What are the finite subgroups of $PGL_2(K)$ whose orders are divisible by $p$? And if $G$ and $H$ are two such subgroups that are isomorphic, can one say when ...