2022/12/03 by Freedman, Saul D. · 1 citation
#05C25 (Secondary) #20E32 (Primary) 20D60 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2212.01616
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 ∖ 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.