Minimal invariable generating sets

TL;DR

This paper investigates the properties and behavior of minimal invariable generating sets in finite groups, focusing on their structure and the conditions under which they generate the entire group.

Contribution

It introduces the concept of minimal invariable generating sets and explores their characteristics and existence within finite groups.

Findings

Characterization of minimal invariable generating sets

Conditions for their existence in finite groups

Insights into their structural properties

Abstract

A subset $S$ of a group $G$ invariably generates $G$ if, when each element of $S$ is replaced by an arbitrary conjugate, the resulting set generates $G.$ An invariable generating set $X$ of $G$ is called minimal if no proper subset of $X$ invariably generates $G.$ We will address several questions related to the behaviour of minimal invariable generating sets of a finite group.