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 $d(G)\le \psi (G)\le d(G)+5$. 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.

23 pages

References in corpus (1)

• Invariable generation and the chebotarev invariant of a finite group

Cited by in corpus (1)

• Generating maximal subgroups of finite almost simple groups