The non-isolated vertices in the generating graph of a direct powers of simple groups |
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Authors: | Eleonora Crestani Andrea Lucchini |
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Institution: | 1. Dipartimento di Matematica, Universitá di Padova, Via Trieste 63, 35121, Padova, Italy
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Abstract: | For a finite group G let Γ(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. Many deep results on the generation of the finite simple groups G can be equivalently stated as theorems that ensure that Γ(G) is a rich graph, with several good properties. In this paper we want to consider Γ(G δ ) where G is a finite non-abelian simple group and G δ is the largest 2-generated power of G, with the aim to investigate whether the good generation properties of G still affect the behaviour of Γ(G δ ). In particular we prove that the graph obtained from Γ(G δ ) by removing the isolated vertices is 1-arc transitive and connected and we investigate the diameter of this graph. Moreover, some intriguing open questions will be introduced and their solutions will be exemplified for $G=\operatorname{Alt}(5)$ . |
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