The non-commuting, non-generating graph of a non-simple group
Algebraic Combinatorics, Volume 6 (2023) no. 5, pp. 1395-1418.

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 $\left[x,y\right]\ne 1$ and $〈x,y〉\ne G$. 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.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.305
Classification: 20D60, 20F16, 05C25
Keywords: non-commuting non-generating graph, soluble groups, generating graph, graphs defined on groups

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)
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Freedman, Saul D. The non-commuting, non-generating graph of a non-simple group. Algebraic Combinatorics, Volume 6 (2023) no. 5, pp. 1395-1418. doi : 10.5802/alco.305. https://alco.centre-mersenne.org/articles/10.5802/alco.305/

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