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.