The non-commuting, non-generating graph of a finite simple group
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866912912098983936 |
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| author | Freedman, Saul D. |
| author_facet | Freedman, Saul D. |
| contents | Let $G$ be a group such that $G/Z(G)$ is finite and simple. The non-commuting, non-generating graph $Ξ(G)$ of $G$ has vertex set $G \setminus Z(G)$, with edges corresponding to pairs of elements that do not commute and do not generate $G$. Complementing our previous investigation of this graph for non-simple groups, we show that $Ξ(G)$ is connected with diameter at most $5$, with smaller upper bounds for certain families of groups. Using these bounds, we then prove that when $G$ is simple, the diameter of the complement of the generating graph of $G$ has a tight upper bound of $4$, with the exception of at most one group with a graph of diameter $5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_01616 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The non-commuting, non-generating graph of a finite simple group Freedman, Saul D. Group Theory 20E32 (Primary) 20D60, 05C25 (Secondary) Let $G$ be a group such that $G/Z(G)$ is finite and simple. The non-commuting, non-generating graph $Ξ(G)$ of $G$ has vertex set $G \setminus Z(G)$, with edges corresponding to pairs of elements that do not commute and do not generate $G$. Complementing our previous investigation of this graph for non-simple groups, we show that $Ξ(G)$ is connected with diameter at most $5$, with smaller upper bounds for certain families of groups. Using these bounds, we then prove that when $G$ is simple, the diameter of the complement of the generating graph of $G$ has a tight upper bound of $4$, with the exception of at most one group with a graph of diameter $5$. |
| title | The non-commuting, non-generating graph of a finite simple group |
| topic | Group Theory 20E32 (Primary) 20D60, 05C25 (Secondary) |
| url | https://arxiv.org/abs/2212.01616 |