Strongly base-two groups

TL;DR

This paper investigates finite groups with trivial Frattini subgroup where almost all faithful transitive actions have base size two, focusing on the classification of strongly base-two groups.

Contribution

It characterizes strongly base-two finite groups with trivial Frattini subgroup, advancing understanding of their permutation representations.

Findings

Most faithful transitive actions have base size two in these groups

Classification of strongly base-two groups with trivial Frattini subgroup

Identification of conditions under which a group is strongly base-two

Abstract

Let $G$ be a finite group, let $H$ be a core-free subgroup and let $b(G,H)$ denote the base size for the action of $G$ on $G/H$. Let $\alpha(G)$ be the number of conjugacy classes of core-free subgroups $H$ of $G$ with $b(G,H) \geqslant 3$. We say that $G$ is a strongly base-two group if $\alpha(G) \leqslant 1$, which means that almost every faithful transitive permutation representation of $G$ has base size $2$. In this paper we study the strongly base-two finite groups with trivial Frattini subgroup.