Average dimension of fixed point spaces with applications

TL;DR

This paper establishes an upper bound on the average dimension of fixed point spaces in finite group modules, generalizing a longstanding conjecture and extending recent theorems with various applications in group theory.

Contribution

It proves a generalized bound on fixed point space dimensions for finite groups acting on modules, resolving a 1966 conjecture and extending recent related results.

Findings

Bound on average fixed point space dimension: at most (1/p) times the module dimension.

Proof of a longstanding conjecture by Neumann and Vaughan-Lee.

Generalizations and improvements of results concerning BFC groups.

Abstract

Let $G$ be a finite group, $F$ a field, and $V$ a finite dimensional $FG$-module such that $G$ has no trivial composition factor on $V$. Then the arithmetic average dimension of the fixed point spaces of elements of $G$ on $V$ is at most $(1/p) \dim V$ where $p$ is the smallest prime divisor of the order of $G$. This answers and generalizes a 1966 conjecture of Neumann which also appeared in a paper of Neumann and Vaughan-Lee and also as a problem in The Kourovka Notebook posted by Vaughan-Lee. Our result also generalizes a recent theorem of Isaacs, Keller, Meierfrankenfeld, and Moret\'o. Various applications are given. For example, another conjecture of Neumann and Vaughan-Lee is proven and some results of Segal and Shalev are improved and/or generalized concerning BFC groups.