An Arad and Fisman's theorem on products of conjugacy classes revisited
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914963505807360 |
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| author | Beltrán, Antonio Felipe, María José Melchor, Carmen |
| author_facet | Beltrán, Antonio Felipe, María José Melchor, Carmen |
| contents | A theorem of Z. Arad and E. Fisman establishes that if $A$ and $B$ are two conjugacy classes of a finite group $G$ such that either $AB=A\cup B$ or $AB=A^{-1} \cup B$, then $G$ cannot be non-abelian simple. We demonstrate that, in fact, $\langle A\rangle = \langle B\rangle$ is solvable, the elements of $A$ and $B$ are $p$-elements for some prime $p$, and $\langle A\rangle $ is $p$-nilpotent. Moreover, under the second assumption, it turns out that $A=B$ and this is the only possible case. This research is done by appealing to recently developed techniques and results that are based on the Classification of Finite Simple Groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_02393 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Arad and Fisman's theorem on products of conjugacy classes revisited Beltrán, Antonio Felipe, María José Melchor, Carmen Group Theory A theorem of Z. Arad and E. Fisman establishes that if $A$ and $B$ are two conjugacy classes of a finite group $G$ such that either $AB=A\cup B$ or $AB=A^{-1} \cup B$, then $G$ cannot be non-abelian simple. We demonstrate that, in fact, $\langle A\rangle = \langle B\rangle$ is solvable, the elements of $A$ and $B$ are $p$-elements for some prime $p$, and $\langle A\rangle $ is $p$-nilpotent. Moreover, under the second assumption, it turns out that $A=B$ and this is the only possible case. This research is done by appealing to recently developed techniques and results that are based on the Classification of Finite Simple Groups. |
| title | An Arad and Fisman's theorem on products of conjugacy classes revisited |
| topic | Group Theory |
| url | https://arxiv.org/abs/2410.02393 |