Some primitive linear groups of prime degree

Abstract

A classical problem in finite group theory dating back to Jordan, Klein, E. H. Moore, Dickson, Blichfeldt etc. is to determine all finite subgroups in $\mathit{SL}(n,\mathit{C})$ up to conjugation for some small values of $n$. This question is important in group theory as well as in the study of quotient singularities. Some results of Blichfeldt when $n=3,4$ were generalized to the case of finite primitive subgroups of $\mathit{SL}(5,\mathit{C})$ and $\mathit{SL}(7,\mathit{C})$ by Brauer and Wales. The purpose of this article is to consider the following case. Let $p$ be any odd prime number and $G$ be a finite primitive subgroup of $\mathit{SL}(p,\mathit{C})$ containing a non-trivial monomial normal subgroup $H$ so that $H$ has a non-scalar diagonal matrix. We will classify all these groups $G$ up to conjugation in $\mathit{SL}(p,\mathit{C})$ by exhibiting the generators of $G$ and representing $G$ as some group extensions. In particular, see the Appendix for a list of these subgroups when $p=5$ or 7.