Right Engel-type subgroups and length parameters of finite groups

TL;DR

This paper investigates the relationship between right Engel-type subgroups generated by elements and the Fitting and nonsoluble length parameters of finite groups, providing new bounds and subgroup inclusions.

Contribution

It extends Baer's theorem to right Engel-type subgroups, establishing bounds on group structure parameters based on these subgroups in both soluble and nonsoluble finite groups.

Findings

In soluble groups, if the Fitting height of R_n(g) is k, then g is in F_{k+1}(G).

In nonsoluble groups, bounds are established for the generalized Fitting height of g.

Results generalize earlier work on left Engel-type subgroups to right Engel-type subgroups.

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.