Right Engel-type subgroups and length parameters of finite groups

Khukhro, Evgeny and Shumyatsky, Pavel and Traustason, Gunnar (2019) Right Engel-type subgroups and length parameters of finite groups. Journal of the Australian Mathematical Society . ISSN 1446-7887

Full content URL: https://doi.org/10.1017/S1446788719000181

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Right Engel-type subgroups and length parameters of finite groups
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Abstract

Let $g$ be an element of a finite group $G$ and let $R_{n}(g)$ be the subgroup generated by all the right Engel values $[g,{}_{n}x]$ over $x\in G$. In the case when $G$ is soluble we prove that if, for some $n$, the Fitting height of $R_{n}(g)$ is equal to $k$, then $g$ belongs to the $(k+1)$th Fitting subgroup $F_{k+1}(G)$. For nonsoluble $G$, it is proved that if, for some $n$, the generalized Fitting height of $R_n(g)$ is equal to $k$, then $g$ belongs to the generalized Fitting subgroup $F^*_{f(k,m)}(G)$ with $f(k,m)$ depending only on $k$ and $m$, where $|g|$ is the product of $m$ primes counting multiplicities. It is also proved that if, for some $n$, the nonsoluble length of $R_n(g)$ is equal to $k$, then $g$ belongs to a normal subgroup whose nonsoluble length is bounded in terms of $k$ and $m$. Earlier similar generalizations of Baer's theorem (which states that an Engel element of a finite group belongs to the Fitting subgroup) were obtained by the first two authors in terms of left Engel-type subgroups.

Keywords:Fitting height, nonsoluble length, generalized Fitting height, finite group, Engel
Subjects:G Mathematical and Computer Sciences > G110 Pure Mathematics
Divisions:College of Science > School of Mathematics and Physics
ID Code:35464
Deposited On:11 Apr 2019 13:56

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