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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 primeorder. The \textit{$2$-closure} of $G$, denoted by $G^{(2),\Omega}$, is thelargest 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 isno $2$-closed elusive group. In this paper, we study the \textit{fixity} ofelusive groups, where the fixity of $G$ is the maximal number of fixed pointsof 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 tothe polycirculant conjecture.

Fixity of elusive groups and the polycirculant conjecture