Groups with a p-element acting with a single non-trivial Jordan block on a simple module in characteristic p

Abstract

Abstract Let V be a vector space over a field of characteristic p. In this paper we complete the classification of all irreducible subgroups G of GL ⁢ ( V ) {\mathrm{GL}(V)} that contain a p-element whose Jordan normal form has exactly one non-trivial block, and possibly multiple trivial blocks. Broadly speaking, such a group acting primitively is a classical group acting on a symmetric power of a natural module, a 7-dimensional orthogonal group acting on the 8-dimensional spin module, a complex reflection group acting on a reflection representation, or one of a small number of other examples, predominantly with a self-centralizing cyclic Sylow p-subgroup.