Paper accepted for publication

The paper by Evgeny Khukhro and Pavel Shumyatsky Engel-type subgroups and length parameters of finite groups has been accepted for publication in Israel Journal of Mathematics. The results of the paper have been obtained in collaboration between Evgeny Khukhro of University of Lincoln and Pavel Shumyatsky of University of Brasilia, with Evgeny’s visits to Brasilia supported by CNPq-Brazil grant within the Brazilian Scientific Mobility Program “Ciências sem Fronteiras”.

Abstract: Let $g$ be an element of a finite group $G$ . For a positive integer $n$ , let $E_n(g)$ be the subgroup generated by all commutators $[...[[x,g],g],\dots ,g]$ over $x\in G$ , where $g$ is repeated $n$ times. By Baer’s theorem, if $E_n(g)=1$ , then $g$ belongs to the Fitting subgroup $F(G)$ . We generalize this theorem in terms of certain length parameters of $E_n(g)$ . For soluble $G$ we prove that if, for some $n$ , the Fitting height of $E_n(g)$ is equal to $k$ , then $g$ belongs to the $(k+1)$ th Fitting subgroup $F_{k+1}(G)$ . For nonsoluble $G$ the results are in terms of nonsoluble length and generalized Fitting height. The generalized Fitting height $h^*(H)$ of a finite group $H$ is the least number $h$ such that $F^*_h(H)=H$ , where $F^*_0(H)=1$ , and $F^*_{i+1}(H)$ is the inverse image of the generalized Fitting subgroup $F^*(H/F^*_{i}(H))$ . Let $m$ be the number of prime factors of $|g|$ counting multiplicities. It is proved that if, for some $n$ , the generalized Fitting height of $E_n(g)$ is equal to $k$ , then $g$ belongs to $F^*_{f(k,m)}(G)$ , where $f(k,m)$ depends only on $k$ and $m$ . The nonsoluble length $\lambda (H)$ of a finite group $H$ is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if $\lambda (E_n(g))=k$ , then $g$ belongs to a normal subgroup whose nonsoluble length is bounded in terms of $k$ and $m$ . We also state conjectures of stronger results independent of $m$ and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.