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

TL;DR

This paper studies the structure of a graph associated with a group, revealing how the group's properties influence the graph's connectivity and diameter, with detailed classifications for certain finite groups.

Contribution

It characterizes the connectedness and diameter of the non-commuting, non-generating graph for groups where $G/Z(G)$ is not simple, including detailed classifications for finite groups with specific graph structures.

Findings

Graph is either connected with diameter ≤ 4, or has isolated vertices and a component with diameter ≤ 4, or two components each of diameter 2.

Finite groups with the graph consisting of two components of diameter 2 are fully described.

The study extends understanding of the relationship between group structure and associated graphs.

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 $\langle x, y \rangle \ne G$. 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.