On recognition of the direct squares of the simple groups with abelian Sylow 2-subgroups
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866913380936187904 |
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| author | Li, Tao Moghaddamfar, A. R. Vasil'ev, Andrey V. Wang, Zhigang |
| author_facet | Li, Tao Moghaddamfar, A. R. Vasil'ev, Andrey V. Wang, Zhigang |
| contents | The spectrum of a group is the set of orders of its elements. Finite groups with the same spectra as the direct squares of the finite simple groups with abelian Sylow 2-subgroups are considered. It is proved that the direct square $J_1\times J_1$ of the sporadic Janko group $J_1$ and the direct squares ${^2}G_2(q)\times{^2}G_2(q)$ of the simple small Ree groups ${^2}G_2(q)$ are uniquely characterized by their spectra in the class of finite groups, while for the direct square $PSL_2(q)\times PSL_2(q)$ of a 2-dimensional simple linear group $PSL_2(q)$, there are always infinitely many groups (even solvable groups) with the same spectra. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_07907 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On recognition of the direct squares of the simple groups with abelian Sylow 2-subgroups Li, Tao Moghaddamfar, A. R. Vasil'ev, Andrey V. Wang, Zhigang Group Theory 20D60, 20D06 The spectrum of a group is the set of orders of its elements. Finite groups with the same spectra as the direct squares of the finite simple groups with abelian Sylow 2-subgroups are considered. It is proved that the direct square $J_1\times J_1$ of the sporadic Janko group $J_1$ and the direct squares ${^2}G_2(q)\times{^2}G_2(q)$ of the simple small Ree groups ${^2}G_2(q)$ are uniquely characterized by their spectra in the class of finite groups, while for the direct square $PSL_2(q)\times PSL_2(q)$ of a 2-dimensional simple linear group $PSL_2(q)$, there are always infinitely many groups (even solvable groups) with the same spectra. |
| title | On recognition of the direct squares of the simple groups with abelian Sylow 2-subgroups |
| topic | Group Theory 20D60, 20D06 |
| url | https://arxiv.org/abs/2312.07907 |