On the length of finite factorized groups

Abstract

The nonsoluble length $\lambda (G)$ of a finite group $G$ is defined as the number of nonsoluble factors in a shortest normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. The generalized Fitting height of a finite group $G$ is the least number $h=h^{\ast }(G)$ such that $F_h^{\ast }(G)=G$, where $F_1^{\ast }(G)=F^{\ast }(G)$ is the generalized Fitting subgroup, and $F_{i+1}^{\ast }(G)$ is the inverse image of $F^{\ast }(G/F_i^{\ast }(G))$. It is proved that if a finite group $G=AB$ is factorized by two subgroups of coprime orders, then the nonsoluble length of $G$ is bounded in terms of the generalized Fitting heights of $A$ and $B$. It is also proved that if, say, $B$ is soluble of derived length $d$, then the generalized Fitting height of $G$ is bounded in terms of $d$ and the generalized Fitting height of $A$.