On the common transversal probability in finite groups
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866929622979969024 |
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| author | Aivazidis, S. Loukaki, M. Mueller, T. W. |
| author_facet | Aivazidis, S. Loukaki, M. Mueller, T. W. |
| contents | Let $G$ be a finite group, and let $H$ be a subgroup of $G$. We compute the probability, denoted by $P_G(H)$, that a left transversal of $H$ in $G$ is also a right transversal, thus a two-sided one. Moreover, we define, and denote by $\mathrm{tp}(G)$, the common transversal probability of $G$ to be the minimum, taken over all subgroups $H$ of $G$, of $P_G(H)$. We prove a number of results regarding the invariant $\mathrm{tp}(G)$, like lower and upper bounds, and possible values it can attain. We also show that $\mathrm{tp}(G)$ determines structural properties of $G$. Finally, several open problems are formulated and discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_00986 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the common transversal probability in finite groups Aivazidis, S. Loukaki, M. Mueller, T. W. Group Theory Let $G$ be a finite group, and let $H$ be a subgroup of $G$. We compute the probability, denoted by $P_G(H)$, that a left transversal of $H$ in $G$ is also a right transversal, thus a two-sided one. Moreover, we define, and denote by $\mathrm{tp}(G)$, the common transversal probability of $G$ to be the minimum, taken over all subgroups $H$ of $G$, of $P_G(H)$. We prove a number of results regarding the invariant $\mathrm{tp}(G)$, like lower and upper bounds, and possible values it can attain. We also show that $\mathrm{tp}(G)$ determines structural properties of $G$. Finally, several open problems are formulated and discussed. |
| title | On the common transversal probability in finite groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2209.00986 |