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

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2024年2月1日

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Tao Li,
Ali Reza Moghaddamfar,
Andrey V. Vasil’ev,
Zhigang Wang

摘要

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 $J_1\times J_1$ $J_{1} \times J_{1}$ of the sporadic Janko group $J_1$ $J_{1}$ and the direct squares ${^2}G_2(q)\times {^2}G_2(q)$ $^{2} G_{2} (q) \times^{2} G_{2} (q)$ of the simple small Ree groups ${^2}G_2(q)$ $^{2} G_{2} (q)$ are uniquely characterized by their spectra in the class of finite groups, while for the direct square $PSL_2(q)\times PSL_2(q)$ $P S L_{2} (q) \times P S L_{2} (q)$ of a 2-dimensional simple linear group $PSL_2(q)$ $P S L_{2} (q)$ , there are always infinitely many groups (even solvable groups) with the same spectra.