On finite totally $2$-closed groups
An abstract group $G$ is called totally $2$-closed if $H=H^{(2),\Omega }$ for any set $\Omega $ with $G\cong H\le \mathrm{Sym}(\Omega )$, where $H^{(2),\Omega }$ is the largest subgroup of $\mathrm{Sym}(\Omega )$ whose orbits on $\Omega \times \Omega $ are the same orbits of $H$. In this paper, we c...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2022-09-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.355/ |
| _version_ | 1797651489949220864 |
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| author | Abdollahi, Alireza Arezoomand, Majid Tracey, Gareth |
| author_facet | Abdollahi, Alireza Arezoomand, Majid Tracey, Gareth |
| author_sort | Abdollahi, Alireza |
| collection | DOAJ |
| description | An abstract group $G$ is called totally $2$-closed if $H=H^{(2),\Omega }$ for any set $\Omega $ with $G\cong H\le \mathrm{Sym}(\Omega )$, where $H^{(2),\Omega }$ is the largest subgroup of $\mathrm{Sym}(\Omega )$ whose orbits on $\Omega \times \Omega $ are the same orbits of $H$. In this paper, we classify the finite soluble totally $2$-closed groups. We also prove that the Fitting subgroup of a totally $2$-closed group is a totally $2$-closed group. Finally, we prove that a finite insoluble totally $2$-closed group $G$ of minimal order with non-trivial Fitting subgroup has shape $Z\cdot X$, with $Z=Z(G)$ cyclic, and $X$ is a finite group with a unique minimal normal subgroup, which is nonabelian. |
| first_indexed | 2024-03-11T16:16:33Z |
| format | Article |
| id | doaj.art-b1ecd228c77e408a95c30aca90dd2d30 |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T16:16:33Z |
| publishDate | 2022-09-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-b1ecd228c77e408a95c30aca90dd2d302023-10-24T14:19:34ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692022-09-01360G91001100810.5802/crmath.35510.5802/crmath.355On finite totally $2$-closed groupsAbdollahi, Alireza0Arezoomand, Majid1Tracey, Gareth2Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan 81746-73441, IranUniversity of Larestan, Larestan 74317-16137, IranSchool of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United KingdomAn abstract group $G$ is called totally $2$-closed if $H=H^{(2),\Omega }$ for any set $\Omega $ with $G\cong H\le \mathrm{Sym}(\Omega )$, where $H^{(2),\Omega }$ is the largest subgroup of $\mathrm{Sym}(\Omega )$ whose orbits on $\Omega \times \Omega $ are the same orbits of $H$. In this paper, we classify the finite soluble totally $2$-closed groups. We also prove that the Fitting subgroup of a totally $2$-closed group is a totally $2$-closed group. Finally, we prove that a finite insoluble totally $2$-closed group $G$ of minimal order with non-trivial Fitting subgroup has shape $Z\cdot X$, with $Z=Z(G)$ cyclic, and $X$ is a finite group with a unique minimal normal subgroup, which is nonabelian.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.355/ |
| spellingShingle | Abdollahi, Alireza Arezoomand, Majid Tracey, Gareth On finite totally $2$-closed groups Comptes Rendus. Mathématique |
| title | On finite totally $2$-closed groups |
| title_full | On finite totally $2$-closed groups |
| title_fullStr | On finite totally $2$-closed groups |
| title_full_unstemmed | On finite totally $2$-closed groups |
| title_short | On finite totally $2$-closed groups |
| title_sort | on finite totally 2 closed groups |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.355/ |
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