Orbits of Sylow subgroups of finite permutation groups

TL;DR

This paper investigates the structure of finite permutation groups with a specific Sylow subgroup property, extending known results to 2-transitive groups and characterizing primitive groups satisfying this property.

Contribution

It generalizes previous work on symmetric and alternating groups to include 2-transitive groups and provides a structural characterization of primitive groups with Property _p.

Findings

Extended Property _p to 2-transitive groups

Provided a structural characterization of primitive groups with Property _p

Identified allowable primes for which the property holds

Abstract

We say that a finite group $G$ acting on a set $\Omega$ has Property $(*)_p$ for a prime $p$ if $P_\omega$ is a Sylow $p$-subgroup of $G_\omega$ for all $\omega\in\Omega$ and Sylow $p$-subgroups $P$ of $G$. Property $(*)_p$ 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 $(*)_p$. 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 $(*)_p$ for an allowable prime $p$.