The non-commuting, non-generating graph of a finite simple group

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1. Verfasser: Freedman, Saul D.
Format: Preprint
Veröffentlicht: 2022
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author Freedman, Saul D.
author_facet Freedman, Saul D.
contents Let $G$ be a group such that $G/Z(G)$ is finite and simple. The non-commuting, non-generating graph $Ξ(G)$ of $G$ has vertex set $G \setminus Z(G)$, with edges corresponding to pairs of elements that do not commute and do not generate $G$. Complementing our previous investigation of this graph for non-simple groups, we show that $Ξ(G)$ is connected with diameter at most $5$, with smaller upper bounds for certain families of groups. Using these bounds, we then prove that when $G$ is simple, the diameter of the complement of the generating graph of $G$ has a tight upper bound of $4$, with the exception of at most one group with a graph of diameter $5$.
format Preprint
id arxiv_https___arxiv_org_abs_2212_01616
institution arXiv
publishDate 2022
record_format arxiv
spellingShingle The non-commuting, non-generating graph of a finite simple group
Freedman, Saul D.
Group Theory
20E32 (Primary) 20D60, 05C25 (Secondary)
Let $G$ be a group such that $G/Z(G)$ is finite and simple. The non-commuting, non-generating graph $Ξ(G)$ of $G$ has vertex set $G \setminus Z(G)$, with edges corresponding to pairs of elements that do not commute and do not generate $G$. Complementing our previous investigation of this graph for non-simple groups, we show that $Ξ(G)$ is connected with diameter at most $5$, with smaller upper bounds for certain families of groups. Using these bounds, we then prove that when $G$ is simple, the diameter of the complement of the generating graph of $G$ has a tight upper bound of $4$, with the exception of at most one group with a graph of diameter $5$.
title The non-commuting, non-generating graph of a finite simple group
topic Group Theory
20E32 (Primary) 20D60, 05C25 (Secondary)
url https://arxiv.org/abs/2212.01616