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

Abstract

Let $G$ be a (finite or infinite) group such that $G/Z(G)$ is not simple. The non-commuting, non-generating graph $\Xi (G)$ of $G$ has vertex set $G\setminus Z(G)$, with vertices $x$ and $y$ adjacent whenever $[x,y]\ne 1$ and $⟨x,y⟩\ne G$. We investigate the relationship between the structure of $G$ and the connectedness and diameter of $\Xi (G)$. In particular, we prove that the graph either: (i) is connected with diameter at most $4$; (ii) consists of isolated vertices and a connected component of diameter at most $4$; or (iii) is the union of two connected components of diameter $2$. We also describe in detail the finite groups with graphs of type (iii). In the companion paper arXiv:2212.01616, we consider the case where $G/Z(G)$ is finite and simple.

arXiv comment25 pages. Minor corrections to the proofs of Lemmas 4.9, 5.6 and 5.7

References in corpus (1)

• The non-commuting, non-generating graph of a nilpotent group

Cited by in corpus (1)

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