Fixity of elusive groups and the polycirculant conjecture

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.