Orbits of Sylow subgroups of finite permutation groups

Abstract

We say that a finite group $G$ acting on a set $\hat{I}\text{c◯}$ has Property $(\ast {)}_p$ for a prime $p$ if $P_{\ddot{I}}‰$ is a Sylow $p$-subgroup of $G_{\ddot{I}}‰$ for all $\ddot{I}‰\in \hat{I}\text{c◯}$ and Sylow $p$-subgroups $P$ of $G$. Property $(\ast {)}_p$ arose in the recent work of Tornier (2018) on local Sylow $p$-subgroups of Burger-Mozes groups, and he determined the values of $p$ for which the alternating group $A_n$ and symmetric group $S_n$ acting on $n$ points has Property $(\ast {)}_p$. In this paper, we extend this result to finite $2$-transitive groups and we give a structural characterisation result for the finite primitive groups that satisfy Property $(\ast {)}_p$ for an allowable prime $p$.

minor corrections