Left 3-Engel elements in groups of exponent 5
Left 3-Engel elements in groups of exponent 5
复制标题
左 3-指数 5 组中的恩格尔元素
DOI：
10.1016/j.jalgebra.2014.05.017
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发表时间：
2014
影响因子：
0.9
通讯作者：
G. Traustason
中科院分区：
文献类型：
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作者：
G. Traustason
群G的左3-Engel元是否总是包含在群G的Hirsch-Plotkin根中，至今仍是一个悬而未决的问题。本文开始对这一问题进行了系统的研究。这个问题首先被改写为说，某种类型的群体是局部幂零的。我们把这些群称为三明治群，因为它们可以被看作是李代数背景下三明治代数的类似物。证明了任意3-生成元夹心群都是幂零群，并得到了自由3-生成元夹心群的幂共轭表示。作为应用，我们证明了任意指数为5的群G的左3-Engel元都在G的Hirsch-Plotkin根中.
It is still an open question whether a left 3-Engel element of a groupGis always contained in the Hirsch–Plotkin radical ofG. In this paper we begin a systematic study of this problem. The problem is first rephrased as saying that a certain type of groups are locally nilpotent. We refer to these groups as sandwich groups as they can be seen as the analogs of sandwich algebras in the context of Lie algebras. We show that any 3-generator sandwich group is nilpotent and obtain a power-conjugation presentation for the free 3-generator sandwich group. As an application we show that the left 3-Engel elements in any groupGof exponent 5 are in the Hirsch–Plotkin radical ofG.