Notes on $B$-groups
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915006889590784 |
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| author | Ponomarenko, Ilia Ryabov, Grigory |
| author_facet | Ponomarenko, Ilia Ryabov, Grigory |
| contents | Following Wielandt, a finite group $G$ is called a $B$-group (Burnside group) if every primitive group containing a regular subgroup isomorphic to $G$ is doubly transitive. Using a method of Schur rings, Wielandt proved that every abelian group of composite order which has at least one cyclic Sylow subgroup is a $B$-group. Since then, other infinite families of $B$-groups were found by the same method. A simple analysis of the proofs of these results shows that in all of them a stronger statement was proved for the group $G$ under consideration: every primitive Schur ring over $G$ is trivial. A finite group $G$ possessing the latter property, we call $BS$-group (Burnside-Schur group). In the present note, we give infinitely many examples of $B$-groups which are not $BS$-groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22998 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Notes on $B$-groups Ponomarenko, Ilia Ryabov, Grigory Group Theory Following Wielandt, a finite group $G$ is called a $B$-group (Burnside group) if every primitive group containing a regular subgroup isomorphic to $G$ is doubly transitive. Using a method of Schur rings, Wielandt proved that every abelian group of composite order which has at least one cyclic Sylow subgroup is a $B$-group. Since then, other infinite families of $B$-groups were found by the same method. A simple analysis of the proofs of these results shows that in all of them a stronger statement was proved for the group $G$ under consideration: every primitive Schur ring over $G$ is trivial. A finite group $G$ possessing the latter property, we call $BS$-group (Burnside-Schur group). In the present note, we give infinitely many examples of $B$-groups which are not $BS$-groups. |
| title | Notes on $B$-groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2410.22998 |