Thomas Michael Keller

Fixed point spaces, primitive character degrees and conjugacy class sizes

Let G be a finite group that acts on a nonzero finite dimensional vector space V over an arbitrary field. Assume that V is completely reducible as a G-module, and that G fixes no nonzero vector of V. We show that some element g ∈ G has a small fixed-point space in V. Specifically, we prove that we can choose g so that dim C V (g) < (1/p)dim V, where p is the smallest prime divisor of |G|.

A NEW APPROACH TO THE k.GV/-PROBLEM

This paper is concerned with the well-known and long-standing k.GV/-problem: If the finite group G acts faithfully and irreducibly on the finite GF . p/-module V and p does not divide the order of G ,i s the number k.GV/ of conjugacy classes of the semidirect product GV bounded above by the order of V ? Over the past two decades, through the work of numerous people, by using deep character theoretic arguments this question has been answered in the affirmative except for p D 5 for which it is still open. In this paper we suggest a new approach to the k.GV/-problem which is independent of most of the previous work on the problem and which is mainly group theoretical. To demonstrate the potential of the new line of attack we use it to solve the k.GV/-problem for solvable G and large p.

Groups with few conjugacy classes

Let G be a finite group, let p be a prime divisor of the order of G and let k ( G ) be the number of conjugacy classes of G . By disregarding at most finitely many non-solvable p -solvable groups G , we have with equality if and only if if is an integer, and C G ( C p ) = C p . This extends earlier work of Hethelyi, Kulshammer, Malle and Keller.

On nilpotent and solvable quotients of primitive groups

Abstract Extending work of Aschbacher and Guralnick on abelian quotients of finite groups, in this paper we show that if G is a primitive permutation group on a set of size n, then any nilpotent quotient of G has order at most nβ and any solvable quotient of G has order at most nα+1, where β = log 32/log 9 and α = (3 log(48) + log(24))/(3 · log(9)).

Inductive Arguments for the Non-coprime k(GV)-Problem

We present some arguments that can be used in an inductive approach to the non-coprime k(GV)-problem.