The spectrum of a finite group is the set of its element orders. Let $q$ be a power of a prime $p$, with $p \geqslant 5$. It is known that any finite group having the same spectrum as the simple symplectic group $PSp_4(q)$ either is isomorphic to an almost simple group with socle $PSp_4(q)$ or can be homomorphically mapped onto an almost simple group $H$ with socle $PSL_2(q^2)$. We prove that the group $H$ cannot coincide with $PSL_2(q^2)$, i.e., $H$ must contain outer automorphisms of its socle.
Keywords: finite group, element order
Received August 15, 2023
Revised September 19, 2023
Accepted September 25, 2023
Funding Agency: This research was carried out within a state task to the Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences (project no. FWNF-2022-0002).
Mariya Aleksandrovna Grechkoseeva, Dr. Phys.-Math. Sci., Sobolev Institute of Mathematics of the Siberian Branch of the RAS, Novosibirsk, 630090 Russia, e-mail: grechkoseeva@gmail.com
Vladislav Maksimovich Rodionov, Novosibirsk State University, Novosibirsk, 630090 Russia, e-mail: v.rodionov@g.nsu.ru
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Cite this article as: M.A. Grechkoseeva, V.M. Rodionov. On finite groups isospectral to $PSp_4(q)$. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 4, pp. 64–69.