On finite groups with exactly one noncommutator
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909799552122880 |
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| author | Skresanov, Saveliy V. |
| author_facet | Skresanov, Saveliy V. |
| contents | An element $x$ of a group $G$ is a commutator if it can be expressed in the form $x = a^{-1}b^{-1}ab$ for some $a, b \in G$. In 2010 MacHale posed the following problem in the Kourovka notebook: does there exist a finite group $G$, with $|G| > 2$, such that there is exactly one element of $G$ which is not a commutator? We answer this question in the affirmative and provide an infinite series of such groups, the smallest group in our construction having size $16609443840$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_17587 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On finite groups with exactly one noncommutator Skresanov, Saveliy V. Group Theory 20F12 (Primary) 20-08 (Secondary) An element $x$ of a group $G$ is a commutator if it can be expressed in the form $x = a^{-1}b^{-1}ab$ for some $a, b \in G$. In 2010 MacHale posed the following problem in the Kourovka notebook: does there exist a finite group $G$, with $|G| > 2$, such that there is exactly one element of $G$ which is not a commutator? We answer this question in the affirmative and provide an infinite series of such groups, the smallest group in our construction having size $16609443840$. |
| title | On finite groups with exactly one noncommutator |
| topic | Group Theory 20F12 (Primary) 20-08 (Secondary) |
| url | https://arxiv.org/abs/2509.17587 |