A note on Engel groups and local nilpotence
A note on Engel groups and local nilpotence
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DOI：
10.1017/s1446788700001324
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发表时间：
1998-02
期刊：
影响因子：
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通讯作者：
R. Burns;Yuri Medvedev
中科院分区：
文献类型：
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作者：
R. Burns;Yuri Medvedev
摘要本文研究了n-Engel群是否局部幂零的问题。虽然这似乎不太可能在一般情况下，它表明，这是一个大类C的情况下，包括所有剩余可解和剩余有限的群体（实际上所有的群体认为在传统的教科书群论）。这是由主要结果得出的：存在只依赖于n的整数c（n），e（n），使得类C中的每个n阶生成的n-Engel群都是幂零类有限e（n）×幂零类≤c（n）和幂零类≤c（n）×幂零类有限e（n）.证明中的关键是一个n-生成的恩格尔群有n-生成的换位子群。
Abstract This paper is concerned with the question of whether n-Engel groups are locally nilpotent. Although this seems unlikely in general, it is shown here that it is the case for the groups in a large class C including all residually soluble and residually finite groups (in fact all groups considered in traditional textbooks on group theory). This follows from the main result that there exist integers c(n), e(n) depending only on n, such that every finitely generated n-Engel group in the class C is both finite-of-exponent-e(n)–by–nilpotent-of-class≤c(n) and nilpotent-of-class≤c(n)–by–finite-of-exponent-e(n). Crucial in the proof is the fact that a finitely generated Engel group has finitely generated commutator subgroup.