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.

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Keywords: non-commuting non-generating graph, soluble groups, generating graph, graphs defined on groups

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@article{ALCO_2023__6_5_1395_0, author = {Freedman, Saul D.}, title = {The non-commuting, non-generating graph of a non-simple group}, journal = {Algebraic Combinatorics}, pages = {1395--1418}, publisher = {The Combinatorics Consortium}, volume = {6}, number = {5}, year = {2023}, doi = {10.5802/alco.305}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.305/} }

TY - JOUR AU - Freedman, Saul D. TI - The non-commuting, non-generating graph of a non-simple group JO - Algebraic Combinatorics PY - 2023 SP - 1395 EP - 1418 VL - 6 IS - 5 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.305/ DO - 10.5802/alco.305 LA - en ID - ALCO_2023__6_5_1395_0 ER -

%0 Journal Article %A Freedman, Saul D. %T The non-commuting, non-generating graph of a non-simple group %J Algebraic Combinatorics %D 2023 %P 1395-1418 %V 6 %N 5 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.305/ %R 10.5802/alco.305 %G en %F ALCO_2023__6_5_1395_0

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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