Length-Type Parameters of Finite Groups with Almost Unipotent Automorphisms¹

Abstract—Let $\alpha $ be an automorphism of a finite group $G$. For a positive integer $n$, let ${{E}_{{G,n}}}(\alpha )$ be the subgroup generated by all commutators $[...[[x,\alpha ],\alpha ], \ldots ,\alpha ]$ in the semidirect product $G\left\langle \alpha \right\rangle $ over $x \in G$, where $\alpha $ is repeated $n$ times. By Baer’s theorem, if ${{E}_{{G,n}}}(\alpha ) = 1$, then the commutator subgroup $[G,\alpha ]$ is nilpotent. We generalize this theorem in terms of certain length parameters of ${{E}_{{G,n}}}(\alpha )$. For soluble $G$ we prove that if, for some $n$, the Fitting height of ${{E}_{{G,n}}}(\alpha )$ is equal to $k$, then the Fitting height of $[G,\alpha ]$ is at most $k + 1$. For nonsoluble $G$ the results are in terms of the 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 the order $|\alpha |$ counting multiplicities. It is proved that if, for some $n$, the generalized Fitting height of ${{E}_{{G,n}}}(\alpha )$ is equal to $k$, then the generalized Fitting height of $[G,\alpha ]$ is bounded in terms of $k$ 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 $\lambda ({{E}_{{G,n}}}(\alpha )) = k$, then the nonsoluble length of $[G,\alpha ]$ 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.