Linear groups with orders having certain large prime divisors

In this paper we obtain a classification of those subgroups of the finite general linear group GL(d)(q) with orders divisible by a primitive prime divisor of q(e) - 1 for some e>1/2d. In the course of the analysis, we obtain new results on modular representations of finite almost simple groups. In particular, in the last section, we obtain substantial extensions of the results of Landazuri and Seitz on small cross-characteristic representations of some of the finite classical groups.