On finitely generated profinite groups, II: products in quasisimple groups
On finitely generated profinite groups, II: products in quasisimple groups
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关于有限生成的有限群，II：拟单群中的乘积
DOI：
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发表时间：
2006
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通讯作者：
D. Segal
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文献类型：
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作者：
N. Nikolov;D. Segal
我们证明了两个结果。(1)存在一个绝对常数D，使得对于任意有限拟单群S，给定S的二维任意自同构，S的每一个元素都等于由给定自同构定义的D ‘扭曲对易子’的积。(2)给定一个自然数q，存在C = C(q)和M = M(q)使得：如果S是一个有限拟单群，具有|S/Z(S)| >C, βj (j =1，…， M)是S的任意自同构，qj (j =1，…， M)是q的任意因数，则存在S的内自同构αj使得S = M [S, (αjβj) q j]。这些结果依赖于有限单群的分类，需要它们来完成第一部分主要定理的证明。
We prove two results. (1) There is an absolute constant D such that for any finite quasisimple group S, given 2D arbitrary automorphisms of S, every element of S is equal to a product of D ‘twisted commutators’ defined by the given automorphisms. (2) Given a natural number q, there exist C = C(q) and M = M(q) such that: if S is a finite quasisimple group with |S/Z(S)| >C , βj (j =1 ,... , M) are any automorphisms of S, and qj (j =1 ,... , M) are any divisors of q, then there exist inner automorphisms αj of S such that S = � M [S, (αjβj) q j ]. These results, which rely on the classification of finite simple groups, are needed to complete the proofs of the main theorems of Part I.