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

Abstract

Let $G$ be a group such that $G/Z(G)$ is finite and simple. The non-commuting, non-generating graph $\Xi (G)$ 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 $\Xi (G)$ 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 $5$.

arXiv comment25 pages. Version 3: In addition to several minor changes, we incorporate new information on the maximal subgroups of the monster group to correct results on that group's non-commuting, non-generating graph; non-generating graph; and intersection graph. To appear in Q. J. Math

References in corpus (5)

• Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type
• The non-commuting, non-generating graph of a nilpotent group
• The intersection graph of a finite simple group has diameter at most 5
• The non-commuting, non-generating graph of a non-simple group
• Explicit construction of the maximal subgroups of the Monster