On the structure of finite groups isospectral to finite simple groups

Abstract

Finite groups are said to be isospectral if they have the same sets of element orders. A finite nonabelian simple group $L$ is said to be almost recognizable by spectrum if every finite group isospectral to $L$ is an almost simple group with socle isomorphic to $L$. It is known that all finite simple sporadic, alternating and exceptional groups of Lie type, except $J_2$, $A_6$, $A_{10}$ and ${}^3{D}_4(2)$, are almost recognizable by spectrum. The present paper is the final step in the proof of the following conjecture due to V.D. Mazurov: there exists a positive integer $d_0$ such that every finite simple classical group of dimension larger than $d_0$ is almost recognizable by spectrum. Namely, we prove that a nonabelian composition factor of a~finite group isospectral to a finite simple symplectic or orthogonal group $L$ of dimension at least 10, is either isomorphic to $L$ or not a group of Lie type in the same characteristic as $L$, and combining this result with earlier work, we deduce that Mazurov's conjecture holds with $d_0=60$.

13 pages

References in corpus (1)

• On finite groups isospectral to simple classical groups

Cited by in corpus (8)

• Criterion of unrecognizability of a finite group by its Gruenberg-Kegel graph
• Finite groups isospectral to simple groups
• On orders of elements of finite almost simple groups with linear or unitary socle
• On the nilpotency of the solvable radical of a finite group isospectral to a simple group
• On recognition of symplectic and orthogonal groups of small dimensions by spectrum
• On the prime graph of a finite group with unique nonabelian composition factor
• The graph of atomic divisors and constructive recognition of finite simple groups
• On recognition of direct powers of finite simple linear groups by spectrum