$p$-Parts of Stabilizers in Primitive Permutation Groups

TL;DR

This paper investigates the structure of primitive permutation groups with order divisible by p^2, showing that solvable groups contain stabilizers with specific p-part properties and providing counting methods for non-solvable cases.

Contribution

It establishes the existence of stabilizers with particular p-part sizes in solvable primitive groups and introduces a counting approach for non-solvable groups.

Findings

Existence of stabilizers with 1<|S|_p<|G|_p in solvable groups

Counting argument applicable to non-solvable groups

Insights into the p-part structure of primitive permutation groups

Abstract

Let G be a primitive permutation group on a finite set Omega. Let p^2 divide |G|, for a prime p. We show that when G is solvable, there exists a subset of Omega whose stabilizer S has the property that 1<|S|_p<|G|_p. We offer a counting argument which should be helpful when G is not solvable.