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| Natura: | Preprint |
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2023
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| Accesso online: | https://arxiv.org/abs/2311.04057 |
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| _version_ | 1866929620154056704 |
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| author | Huang, Hong Yi Li, Cai Heng Zhu, Yan Zhou |
| author_facet | Huang, Hong Yi Li, Cai Heng Zhu, Yan Zhou |
| contents | The classification of the finite primitive permutation groups of rank $3$ was completed in the 1980s and this landmark achievement has found a wide range of applications. In the general transitive setting, a classical result of Higman shows that every finite imprimitive rank $3$ permutation group $G$ has a unique non-trivial block system $\mathcal{B}$ and this provides a natural way to partition the analysis of these groups. Indeed, the induced permutation group $G^{\mathcal{B}}$ is $2$-transitive and one can also show that the action induced on each block in $\mathcal{B}$ is also $2$-transitive (and so both induced groups are either affine or almost simple). In this paper, we make progress towards a classification of the rank $3$ imprimitive groups by studying the case where the induced action of $G$ on a block in $\mathcal{B}$ is of affine type. Our main theorem divides these rank $3$ groups into four classes, which are defined in terms of the kernel of the action of $G$ on $\mathcal{B}$. In particular, we completely determine the rank $3$ semiprimitive groups for which $G^{\mathcal{B}}$ is almost simple, extending recent work of Baykalov, Devillers and Praeger. We also prove that if $G$ is rank $3$ semiprimitive and $G^{\mathcal{B}}$ is affine, then $G$ has a regular normal subgroup which is a special $p$-group for some prime $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_04057 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On finite permutation groups of rank three Huang, Hong Yi Li, Cai Heng Zhu, Yan Zhou Group Theory The classification of the finite primitive permutation groups of rank $3$ was completed in the 1980s and this landmark achievement has found a wide range of applications. In the general transitive setting, a classical result of Higman shows that every finite imprimitive rank $3$ permutation group $G$ has a unique non-trivial block system $\mathcal{B}$ and this provides a natural way to partition the analysis of these groups. Indeed, the induced permutation group $G^{\mathcal{B}}$ is $2$-transitive and one can also show that the action induced on each block in $\mathcal{B}$ is also $2$-transitive (and so both induced groups are either affine or almost simple). In this paper, we make progress towards a classification of the rank $3$ imprimitive groups by studying the case where the induced action of $G$ on a block in $\mathcal{B}$ is of affine type. Our main theorem divides these rank $3$ groups into four classes, which are defined in terms of the kernel of the action of $G$ on $\mathcal{B}$. In particular, we completely determine the rank $3$ semiprimitive groups for which $G^{\mathcal{B}}$ is almost simple, extending recent work of Baykalov, Devillers and Praeger. We also prove that if $G$ is rank $3$ semiprimitive and $G^{\mathcal{B}}$ is affine, then $G$ has a regular normal subgroup which is a special $p$-group for some prime $p$. |
| title | On finite permutation groups of rank three |
| topic | Group Theory |
| url | https://arxiv.org/abs/2311.04057 |