可数范畴Engel群的李方法：4-Engel 5-群的Wilson猜想

中文摘要 AI 辅助

本文关注Wilson在1981年提出的如下猜想：每个局部幂零的可数范畴群都是幂零的。在前期关于该猜想的李代数类比工作基础上，我们运用李方法推导得出Wilson猜想在n-Engel群中的首个非平凡情形：可数范畴的3-Engel群与4-Engel 5-群是幂零的，由此还可推出奇指数的可数范畴4-Engel群是幂零的。这一结论通过Lazard对应的特殊情形实现，借助计算机代数系统验证。我们还研究了群、李代数、结合代数三类范畴间的幂零性结果转移问题，其中证明了Wilson猜想蕴含结合代数对应的幂零性结论（交换情形除外）。

英文摘要

We are interested in the following conjecture of Wilson from 1981: every locally nilpotent countably categorical group is nilpotent. Following our previous work on the Lie algebra analogue of the conjecture, we use Lie methods to deduce the first nontrivial cases of the Wilson conjecture for n-Engel groups: countably categorical 3-Engel groups and 4-Engel 5-groups are nilpotent, from which we also conclude that countably categorical 4-Engel groups of odd exponent are nilpotent. This is implemented via an exceptional case of the Lazard correspondence, checked using computer algebra systems. We also study the transfer of nilpotency results between the three categories: groups, Lie algebras, associative algebras. Among other things, we prove that the Wilson conjecture implies the analogous nilpotency statement for associative algebras (modulo the commutative case).

发表机构

• University of the Basque Country（巴斯克大学）

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