关于有限群与有限单群等谱的结构
On the structure of finite groups isospectral to finite simple groups
摘要：
有限群称为等谱群，如果它们具有相同的元素阶集。有限非交换单群$L$称为几乎可谱群，如果每一个与$L$等谱的有限群都是几乎单群，其基柱同构于$L$ .已知除$J_2$，$A_6$，$A_{10}$和$^3D_4(2)$外，所有有限的李型散生、交错和例外单群几乎都是谱可识别的.本文是证明V.D.马祖罗夫：存在一个正整数$d_0$，使得维数大于$d_0$的有限单经典群几乎可由谱识别.也就是说，我们证明了一个与维数至少为10的有限单辛群或正交群$L$等谱的~有限群的非交换合成因子要么同构于$L$，要么不是一个具有与$L$相同特征的Lie型群，并将这个结果与以前的工作相结合，我们推出Mazurov猜想在$d_0=60$中成立.
英文摘要：
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 $^3D_4(2)$, 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 $d_0=60$.
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