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

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.