Robert M. Guralnick

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 GLd (q) with orders divisible by a primitive prime divisor of qe − 1 for some . 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. 1991 Mathematics Subject Classification: primary 20G40; secondary 20C20, 20C33, 20C34, 20E99.

Schur covers and Carlitz's conjecture

AbstractWe use the classification of finite simple groups and covering theory in positive characteristic to solve Carlitz’s conjecture (1966). An exceptional polynomialf over a finite field

A note on commuting pairs of matrices

{\mathbb{F}}_q

Simple groups admit Beauville structures

is a polynomial that is a permutation polynomial on infinitely many finite extensions of

Low-Dimensional Representations of Special Linear Groups in Cross Characteristics

{\mathbb{F}}_q

Products of conjugacy classes and fixed point spaces

. Carlitz’s conjecture saysf must be of odd degree (ifq is odd). Indeed, excluding characteristic 2 and 3, arithmetic monodromy groups of exceptional polynomials must be affine groups.We don’t, however, know which affine groups appear as the geometric metric monodromy group of exceptional polynomials. Thus, there remain unsolved problems. Riemann’s existence theorem in positive characteristic will surely play a role in their solution. We have, however, completely classified the exceptional polynomials of degree equal to the characteristic. This solves a problem from Dickson’s thesis (1896). Further, we generalize Dickson’s problem to include a description of all known exceptional polynomials.Finally: The methods allow us to consider coversX→