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

TL;DR

This paper resolves Peter Neumann’s 1966 conjecture by proving that every finite primitive non-regular permutation group of degree $n$ contains an element fixing between $1$ and $n^{1/3}-1$ points, a bound that is sharp in the affine case and nearly optimal otherwise. The authors reduce to the almost-simple case via the O'Nan–Scott theorem and then tackle almost-simple groups by exhaustively analyzing alternating, sporadic, exceptional, and classical families, using a blend of centralizer estimates, representation theory, and subgroup structure. A key methodological thrust is bounding the Fixed-Point Ratio via $|G:M|^{1/3}$, combining CFSG-inspired subgroup classifications with Green correspondence, invariable generation, and detailed case analyses across Lie type and alternating groups. The results also yield density-type corollaries (via Frobenius density) connecting fixed-point behavior to reductions modulo primes and to irreducible polynomials, highlighting the interplay between permutation group theory and number-theoretic density phenomena. Overall, the work completes the strong Neumann conjecture by delivering a robust, uniformly bound fixed-point framework across all primitive non-regular finite groups and across the CFSG-informed landscape of finite simple 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.

Theorems & Definitions (141)

• Conjecture 1.1: P. Neumann, 1966
• Theorem 1.2
• Corollary 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4