Jan Saxl

A classification of the maximal subgroups of the finite alternating and symmetric groups

Following the classification of finite simple groups, one of the major problems in finite group theory today is the determination of the maximal subgroups of the almost simple groups-that is, of groups X such that X0 u X< Aut X, for some finite non-abelian simple group X0. The problem has been solved for most sporadic groups and groups of Lie type of low rank (see [ 19, Sects. 3, 43 for discussion and references). This paper is a contribution to the case where X0 is an alternating group A, (so that for n # 6, X is A, or S,). The maximal subgroups of A, and S, are known for several classes of degrees n:

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

Logical stability in group theory

{\mathbb{F}}_q

The rational function analogue of a question of Schur and exceptionality of permutation representations

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

Subgroups of Algebraic Groups Containing Regular Unipotent Elements

{\mathbb{F}}_q