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 SL(n,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 SL(5,C) and SL(7,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 SL(p,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 SL(p,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.