The generating graph of finite soluble groups

Abstract

For a finite group G, let Gamma(G) denote the graph defined on the non-identity elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. We prove that if G is soluble, then the non-isolated vertices of Gamma(G) belong to a unique connected component

The Generating Graph of Infinite Abelian Groups

“…Proof. The statement is clearly true if γ is of kind (1) or (2). In the third case b 1 = a 1 · a 2 , so b 1 , a 1 = a 1 , a 2 = F.…”

“…Let Γ * (G) be the subgraph of Γ(G) that is induced by all the vertices that are not isolated. In [2] it is proved that if G is a 2-generated finite soluble group, then Γ * (G) is connected. Later on in [4] the second author investigated the diameter of such a graph proving that if G is a finite soluble group then diam(Γ * (G)) ≤ 3.…”

Generating sets of finite groups

“…Such groups G have long been studied by means of the generating graph, whose vertices are the elements of G, the edges being the 2-element generating sets. The generating graph was defined by Liebeck and Shalev in [16], and has been further investigated by many authors: see for example [3,5,6,12,18,19,20,23] for some of the range of questions that have been considered. Many deep structural results about finite groups can be expressed in terms of the generating graph.…”

The diameter of the generating graph of a finite soluble group

“…Let ∆(G) be the subgraph of Γ(G) that is induced by all of the vertices that are not isolated. In [4] it is proved that if G is a 2-generated soluble group, then ∆(G) is connected. In this paper we investigate the diameter diam(∆(G)) of this graph.…”

“…It follows from the proof of [4,Lemma 6] that the number, say φ G,N (X, k), of k-tuples (g 1 n 1 , . .…”