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A Criterion for Nonsolvability of a Finite Group and Recognition of Direct Squares of Simple Groups

Algebra and Logic Aims and scope

The spectrum ω(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 ω(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 ⩾ 8 and q ≠ 32, the direct square Sz(q) × 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) × Sz(32), there are exactly four finite groups with the same spectrum.

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Acknowledgments

The authors would like to express their deep gratitude and admiration to V. D. Mazurov whose article [6] served as a source of inspiration for this paper. Thanks also are due to A. A. Buturlakin and the reviewer for valuable comments.

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Authors and Affiliations

  1. School of Science, Hainan University, Haikou, Hainan, P.R. China

    Zh. Wang & A. V. Vasil’ev

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    A. V. Vasil’ev & M. A. Grechkoseeva

  3. Kabardino-Balkarian State University, Nalchik, Russia

    A. Kh. Zhurtov

Authors
  1. Zh. Wang
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  2. A. V. Vasil’ev
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  3. M. A. Grechkoseeva
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  4. A. Kh. Zhurtov
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Correspondence to Zh. Wang.

Additional information

Translated from Algebra i Logika, Vol. 61, No. 4, pp. 424-442, July-August, 2022. Russian https://doi.org/10.33048/alglog.2022.61.403.

Dedicated to V. D. Mazurov on the occasion of his 80th birthday

Supported by the National Natural Science Foundation of China (NSFC), grant No. 12171126. (A. V. Vasil’ev)

Supported by the Program of Fundamental Research RAS, project FWNF-2022-0002. (A. V. Vasil’ev and M. A. Grechkoseeva)

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Wang, Z., Vasil’ev, A.V., Grechkoseeva, M.A. et al. A Criterion for Nonsolvability of a Finite Group and Recognition of Direct Squares of Simple Groups. Algebra Logic 61, 288–300 (2022). https://doi.org/10.1007/s10469-023-09697-z

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  • DOI: https://doi.org/10.1007/s10469-023-09697-z

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