On minimal degree of transitive permutation groups with stabiliser being a $2$-group

TL;DR

This paper establishes a lower bound of two-thirds of the degree for the minimal degree of certain transitive permutation groups with specific stabiliser properties, using the classification of finite simple groups.

Contribution

It proves a new lower bound on the minimal degree for transitive groups with stabilisers as 2-groups, under the condition of having no non-trivial normal 2-subgroups.

Findings

Minimal degree is at least 2/3 of the degree for the specified groups.

The proof relies on the classification of finite simple groups.

Provides structural insights into permutation groups with 2-group stabilisers.

Abstract

The minimal degree of a permutation group $G$ is defined as the minimal number of non-fixed points of a non-trivial element of $G$. In this paper we show that if $G$ is a transitive permutation group of degree $n$ having no non-trivial normal $2$-subgroups such that the stabiliser of a point is a $2$-group, then the minimal degree of $G$ is at least $\frac{2}{3}n$. The proof depends on the classification of finite simple groups.