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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2026. Vol 55

𝑁-recognizability of Groups 𝐴𝑙𝑡𝑝 × 𝐴𝑙𝑡5, Where 𝑝 > 1361 Is a Prime Number

Author(s)

Ilya B. Gorshkov1,3, Vitalii D. Shepelev1,2

1 Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russian Federation 

2 Novosibirsk State University, Novosibirsk, Russian Federation  

3 Siberian Federal University, Krasnoyarsk, Russian Federation

Abstract
Given a finite group 𝐿, let 𝑁(𝐿) denote the set of its conjugacy class sizes. Let 𝑋 and 𝑌 be sets of natural numbers, 𝐺 be a finite group such that 𝑁(𝐺) = 𝑋 × 𝑌 . In the article [16] the question is formulated: for which sets 𝑋 and 𝑌 is it true that 𝐺 ≃ 𝐴 × 𝐵, where 𝑁(𝐴) = 𝑋 and 𝑁(𝐵) = 𝑌 ? More than 30 years ago, J. Thompson formulated a conjecture that any finite simple group is uniquely determined by its set of sizes of conjugacy classes in the class of finite groups with trivial center. In 2019, the validity of this conjecture was proven. In 2020, it was noted that in addition to simple groups, some direct products of simple groups are also determined by this set. We prove that if 𝑁(𝐺) = 𝑁(𝐴𝑙𝑡𝑝 × 𝐴𝑙𝑡5), where 𝑝 is a prime greater than 1361 and the group 𝐺 has a trivial center, then 𝐺 ≃ 𝐴𝑙𝑡5 × 𝐴𝑙𝑡𝑝.
About the Authors

Ilya B. Gorshkov, Dr. Sci. (Phys.-Math.), Sobolev Institute of Mathematics SB RAS, Novosibirsk, 630090, Russian Federation; Siberian Federal University, Krasnoyarsk, 660041, Russian Federation, ilygor8@gmail.com

Vitalii D. Shepelev, Sobolev Institute of Mathematics SB RAS, Novosibirsk, 630090, Russian Federation; Novosibirsk State University, Novosibirsk, 630090, Russian Federation, v.shepelev@g.nsu.ru

For citation
Gorshkov I. B., Shepelev V. D. 𝑁-recognizability of Groups 𝐴𝑙𝑡𝑝 × 𝐴𝑙𝑡5, Where 𝑝 > 1361 Is a Prime Number. The Bulletin of Irkutsk State University. Series Mathematics, 2026, vol. 55, pp. 110–122. https://doi.org/10.26516/1997-7670.2026.55.110
Keywords
finite groups, conjugacy class, alternating groups
UDC
512.542
MSC
20B05, 20B30
DOI
https://doi.org/10.26516/1997-7670.2026.55.110
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