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On recognition of the direct squares of the simple groups with abelian Sylow 2-subgroups

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Abstract

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.

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Acknowledgements

The authors are very grateful to Maria Grechkoseeva for useful comments and fruitful discussions. A. V. Vasil’ev was supported by RAS Fundamental Research Program, project FWNF-2022-0002, and by National Natural Science Foundation of China (No. 12171126).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Hainan University, Haikou, Hainan, People’s Republic of China

    Tao Li & Zhigang Wang

  2. Faculty of Mathematics, K. N. Toosi University of Technology, P. O. Box 16765-3381, Tehran, Iran

    Ali Reza Moghaddamfar

  3. Sobolev Institute of Mathematics, Novosibirsk, Russia, 630090

    Andrey V. Vasil’ev

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  1. Tao Li
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  2. Ali Reza Moghaddamfar
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  3. Andrey V. Vasil’ev
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  4. Zhigang Wang
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Correspondence to Andrey V. Vasil’ev.

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Dedicated to Professor Shi Wujie on the occasion of his 80th birthday.

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Li, T., Moghaddamfar, A.R., Vasil’ev, A.V. et al. On recognition of the direct squares of the simple groups with abelian Sylow 2-subgroups. Ricerche mat (2024). https://doi.org/10.1007/s11587-024-00847-8

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