On finite groups isospectral to simple classical groups

Abstract

The spectrum $\omega (G)$ of a finite group $G$ is the set of element orders of $G$. Finite groups $G$ and $H$ are isospectral if their spectra coincide. Suppose that $L$ is a simple classical group of sufficiently large dimension (the lower bound varies for different types of groups but is at most 62) defined over a finite field of characteristic $p$. It is proved that a finite group $G$ isospectral to $L$ cannot have a nonabelian composition factor which is a group of Lie type defined over a field of characteristic distinct from $p$. Together with a series of previous results this implies that every finite group $G$ isospectral to $L$ is `close' to $L$. Namely, if $L$ is a linear or unitary group, then $L⩽G⩽\mathrm{Aut}(L)$, in particular, there are only finitely many such groups $G$ for given $L$. If $L$ is a symplectic or orthogonal group, then $G$ has a unique nonabelian composition factor $S$ and, for given $L$, there are at most 3 variants for $S$ (including $S\simeq L$).

70 pages

References in corpus (2)

• An adjacency criterion for the prime graph of a finite simple group
• Cocliques of maximal size in the prime graph of a finite simple group

Cited by in corpus (16)

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• On groups having the prime graph as alternating and symmetric groups
• Criterion of nonsolvability of a finite group and recognition of direct squares of simple groups
• On existence of normal p-complement of finite groups with restrictions on the conjugacy class sizes
• Almost cyclic elements in cross-characteristic representations of finite groups of Lie type
• Simple Groups Whose Prime Graph or Solvable Graph is Split
• The graph of atomic divisors and constructive recognition of finite simple groups
• The group $J_4\times J_4$ is recognizable by spectrum
• On Thompson's conjecture for alternating group of large degree
• On recognition of direct powers of finite simple linear groups by spectrum
• On recognition of the direct squares of the simple groups with abelian Sylow 2-subgroups