Modular analogues of Jordan's theorem for finite linear groups

TL;DR

This paper extends Jordan's theorem to finite linear groups over algebraically closed fields of positive characteristic, establishing bounds on normal abelian subgroups and providing optimal bounds for all degrees and characteristics.

Contribution

It provides the first comprehensive analysis of Jordan-type bounds for finite linear groups in positive characteristic, including optimal bounds for all degrees and characteristics.

Findings

Bounds similar to characteristic 0 case are established.

Complete characterization of optimal bounds for all degrees and characteristics.

Normal subgroups of bounded index are identified for finite linear groups.

Abstract

In 1878, Jordan showed that a finite subgroup of GL(n,C) contains an abelian normal subgroup whose index is bounded by a function of n alone. Previously, the author has given precise bounds. Here, we consider analogues for finite linear groups over algebraically closed fields of positive characteristic l. A larger normal subgroup must be taken, to eliminate unipotent subgroups and groups of Lie type and characteristic l, and we show that generically the bound is similar to that in characteristic 0 - being (n+1)!, or (n+2)! when l divides (n+2) - given by the faithful representations of minimal degree of the symmetric groups. A complete answer for the optimal bounds is given for all degrees n and every characteristic l.