Generating sets of finite 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.