The generating graph of a profinite group

Abstract

We consider the graph Γ virt (G) whose vertices are the elements of a finitely generated profinite group G and where two vertices x and y are adjacent if and only if they topologically generate an open subgroup of G. We investigate the connectivity of the graph ∆ virt (G) obtained from Γ virt (G) by removing its isolated vertices. In particular we prove that for every positive integer t, there exists a finitely generated prosoluble group G with the property that ∆ virt (G) has precisely t connected components. Moreover we study the graphΓ virt (G), whose vertices are again the elements of G and where two vertices are adjacent if and only if there exists a minimal generating set of G containing them. In this case we prove that the subgraph∆ virt (G) obtained removing the isolated vertices is connected and has diameter at most 3.