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Mat. Zametki, 2025 Volume 117, Issue 4, Pages 494–504 (Mi mzm14506)

Recognizability of the groups $PSp_8(7^m)$ by the set of element orders

M. A. Grechkoseeva

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

Abstract: The set of orders of the elements of a finite group $G$ is called the spectrum of $G$. A group $G$ is said to be unrecognizable by spectrum if there are infinitely many pairwise nonisomorphic finite groups that have the same spectrum as $G$. There is a conjecture that every finite simple classical group unrecognizable by spectrum is contained in the following list: $PSL_3(3)$, $PSU_3(q)$, $PSU_5(2)$, $PSp_4(q)$, $PSp_8(q)$, and $P\Omega_9(q)$. The only groups in this list that were not known to be unrecognizable by spectrum are $PSp_8(7^m)$. In the present paper, it is shown that $PSp_8(7^m)$ are not unrecognizable and, moreover, any of these groups is uniquely (up to isomorphism) determined by its spectrum in the class of all finite groups.

Keywords: orders of elements, simple group, symplectic group, orthogonal group, representations in defining characteristic.

UDC: 512.542

MSC: 20D06, 20D60

Received: 10.06.2024

DOI: 10.4213/mzm14506


 English version:
Mathematical Notes, 2025, 117:4, 538–546


© Steklov Math. Inst. of RAS, 2025