Fixity of elusive groups and the polycirculant conjecture

TL;DR

This paper investigates elusive groups in permutation group theory, focusing on their fixity and providing partial evidence supporting the polycirculant conjecture by showing no such groups are solvable with fixity at most 5.

Contribution

It introduces the concept of fixity for elusive groups and proves that no 2-closed elusive solvable groups have fixity at most 5, advancing understanding of the polycirculant conjecture.

Findings

No 2-closed elusive solvable groups with fixity ≤ 5 exist.

Provides partial evidence for the polycirculant conjecture.

Analyzes the structure of elusive groups in the context of fixity.

Abstract

Let $G\leq{\rm Sym}(\Omega)$ be transitive. Then $G$ is called \textit{elusive} on $\Omega$ if it has no fixed point free element of prime order. The \textit{$2$-closure} of $G$, denoted by $G^{(2),\Omega}$, is the largest subgroup of ${\rm Sym}(\Omega)$ whose orbits on $\Omega\times\Omega$ are the same orbits of $G$. $G$ is called $2$-closed on $\Omega$ if $G=G^{(2),\Omega}$. The \textit{polycirculant conjecture} states that there is no $2$-closed elusive group. In this paper, we study the \textit{fixity} of elusive groups, where the fixity of $G$ is the maximal number of fixed points of a non-trivial element of $G$. In particular, we prove that there is no $2$-closed elusive solvable group of fixity at most $5$, a partial answer to the polycirculant conjecture.