Group Laws Implying Virtual Nilpotence
Group Laws Implying Virtual Nilpotence
复制标题
DOI：
10.1017/s1446788700003335
复制
发表时间：
2003-06
影响因子：
0.7
通讯作者：
Robert G. Burns;Yuri Medvedev
中科院分区：
文献类型：
--
作者：
Robert G. Burns;Yuri Medvedev
如果ω≡1是一个群律，它隐含在满足它的每个有限生成亚交换群中的虚幂零，那么它蕴含着包括所有剩余或局部可解或有限群在内的一大类群的有限生成群的虚幂零.事实上，满足这样一个定律的群都是有限幂零指数，其中所讨论的幂零类和指数都在上面仅就ω的长度有界。这就产生了词语的二分法。最后，如果定律ω≡1满足特定的附加条件--特别是对于任何么一定律或恩格尔定律--那么结论扩展到由所有“局部分级”群组成的更大的类。
Abstract If ω ≡ 1 is a group law implying virtual nilpotence in every finitely generated metabelian group satisfying it, then it implies virtual nilpotence for the finitely generated groups of a large class of groups including all residually or locally soluble-or-finite groups. In fact the groups of satisfying such a law are all nilpotent-by-finite exponent where the nilpotency class and exponent in question are both bounded above in terms of the length of ω alone. This yields a dichotomy for words. Finally, if the law ω ≡ 1 satisfies a certain additional condition—obtaining in particular for any monoidal or Engel law—then the conclusion extends to the much larger class consisting of all ‘locally graded’ groups.