On the connectivity of the generating and rank graphs of finite groups

Abstract

The generating graph encodes how generating pairs are spread among the elements of a group. For more than ten years it has been conjectured that this graph is connected for every finite group. In this paper, we give evidence supporting this conjecture: we prove that it holds for all but a finite number of almost simple groups and give a reduction to groups without non-trivial soluble normal subgroups. Let $d(G)$ be the minimal cardinality of a generating set for $G$. When $d(G)\geq3$, the generating graph is empty and the conjecture is trivially true. We consider it in the more general setting of the rank graph, which encodes how pairs of elements belonging to generating sets of minimal cardinality spread among the elements of a group. It carries information even when $d(G)\geq3$ and corresponds to the generating graph when $d(G)=2$. We prove that it is connected whenever $d(G)\geq3$, giving tools and ideas that may be used to address the original conjecture.