Generating sets of finite groups

TL;DR

This paper introduces a new invariant for finite groups based on element substitution in generating sets, classifies soluble groups by this invariant, and explores automorphisms of their generating graphs.

Contribution

It defines a novel equivalence relation on group elements related to generating sets and characterizes this invariant for soluble and insoluble groups.

Findings

For soluble groups, the invariant is either the minimal number of generators or one more.

For insoluble groups, the invariant is within five of the minimal number of generators.

The invariant helps compute automorphism groups of generating graphs for soluble groups.

Abstract

We investigate the extent to which the exchange relation holds in finite groups $G$. We define a new equivalence relation $\equiv_{\mathrm{m}}$, where two elements are equivalent if each can be substituted for the other in any generating set for $G$. We then refine this to a new sequence $\equiv_{\mathrm{m}}^{(r)}$ of equivalence relations by saying that $x \equiv_{\mathrm{m}}^{(r)}y$ if each can be substituted for the other in any $r$-element generating set. The relations $\equiv_{\mathrm{m}}^{(r)}$ become finer as $r$ increases, and we define a new group invariant $\psi(G)$ to be the value of $r$ at which they stabilise to $\equiv_{\mathrm{m}}$. Remarkably, we are able to prove that if $G$ is soluble then $\psi(G) \in \{d(G), d(G) +1\}$, where $d(G)$ is the minimum number of generators of $G$, and to classify the finite soluble groups $G$ for which $\psi(G) = d(G)$. For insoluble $G$, we show that $d(G) \leq \psi(G) \leq d(G) + 5$. However, we know of no examples of groups $G$ for which $\psi(G) > d(G) + 1$. As an application, we look at the generating graph of $G$, whose vertices are the elements of $G$, the edges being the $2$-element generating sets. Our relation $\equiv_{\mathrm{m}}^{(2)}$ enables us to calculate $\mathrm{Aut}(\Gamma(G))$ for all soluble groups $G$ of nonzero spread, and give detailed structural information about $\mathrm{Aut}(\Gamma(G))$ in the insoluble case.