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

Freedman, Saul D.^1, 2

¹ School of Mathematics and Statistics University of St Andrews St Andrews KY16 9SS (UK)
² Current address: Centre for the Mathematics of Symmetry and Computation The University of Western Australia Crawley, WA 6009 (Australia)

Algebraic Combinatorics, Tome 6 (2023) no. 5, pp. 1395-1418

Résumé

Let $G$ be a (finite or infinite) group such that $G/Z\left(G\right)$ is not simple. The non-commuting, non-generating graph $\Xi \left(G\right)$ of $G$ has vertex set $G\setminus Z\left(G\right)$, 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 $\Xi \left(G\right)$. 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 [17], we consider the case where $G/Z\left(G\right)$ is finite and simple.

Détail

Reçu le : 2022-11-16
Révisé le : 2023-03-05
Accepté le : 2023-03-07
Publié le : 2023-11-07

DOI : 10.5802/alco.305

Classification : 20D60, 20F16, 05C25
Keywords: non-commuting non-generating graph, soluble groups, generating graph, graphs defined on groups

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits