Average dimension of fixed point spaces with applications
Robert M. Guralnick, Attila Maroti

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 be a finite group, a field, and a finite dimensional -module such that has no trivial composition factor on . Then the arithmetic average dimension of the fixed point spaces of elements of on is at most where is the smallest prime divisor of the order of . 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.
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Taxonomy
TopicsFinite Group Theory Research · Coding theory and cryptography · Rings, Modules, and Algebras
