Geodesic
A criterion for nonsolvability of a finite group and recognition of direct squares of simple groups
Algebra i logika, Tome 61 (2022) no. 4, pp. 424-442

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The spectrum $\omega(G)$ of a finite group $G$ is the set of orders of its elements. The following sufficient criterion of nonsolvability is proved: if, among the prime divisors of the order of a group $G$, there are four different primes such that $\omega(G)$ contains all their pairwise products but not a product of any three of these numbers, then $G$ is nonsolvable. Using this result, we show that for $q\geqslant 8$ and $q\neq 32$, the direct square $Sz(q)\times Sz(q)$ of the simple exceptional Suzuki group $Sz(q)$ is uniquely characterized by its spectrum in the class of finite groups, while for $Sz(32)\times Sz(32)$, there are exactly four finite groups with the same spectrum.
Keywords: criterion of nonsolvability, element orders, recognition by spectrum.
Mots-clés : simple exceptional group 
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     author = {Zh. Wang and A. V. Vasil'ev and M. A. Grechkoseeva and A. Kh. Zhurtov},
     title = {A criterion for nonsolvability of a finite group and recognition of direct squares of simple groups},
     journal = {Algebra i logika},
     pages = {424--442},
     publisher = {mathdoc},
     volume = {61},
     number = {4},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2022_61_4_a2/}
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Zh. Wang; A. V. Vasil'ev; M. A. Grechkoseeva; A. Kh. Zhurtov. A criterion for nonsolvability of a finite group and recognition of direct squares of simple groups. Algebra i logika, Tome 61 (2022) no. 4, pp. 424-442. http://geodesic.mathdoc.fr/item/AL_2022_61_4_a2/