On a conjecture of Peter Neumann on fixed points in permutation groups

TL;DR

This paper proves a strengthened version of Peter Neumann's 1966 conjecture, showing that non-regular primitive permutation groups of degree n contain elements fixing between 1 and n^{1/3} points, with this bound being optimal.

Contribution

The paper confirms Neumann's conjecture for non-affine primitive groups, improving the bound from n^{1/2} to n^{1/3} and establishing its optimality.

Findings

Proved the conjecture for non-affine primitive groups.

Established the n^{1/3} bound as best possible.

Extended previous results from affine to non-affine groups.

Abstract

We prove a conjecture of Peter Neumann from 1966, predicting that every finite non-regular primitive permutation group of degree $n$ contains an element fixing at least one point and at most $n^{1/2}$ points. In fact, we prove a stronger version, where $n^{1/2}$ is replaced by $n^{1/3}$, and this is best possible. The case where $G$ is affine was proved by Guralnick and Malle; in this paper we address the case where $G$ is non-affine.