On the length of finite factorized groups

Khukhro, E. I. and Shumyatsky, P. (2015) On the length of finite factorized groups. [MIMS Preprint]

The nonsoluble length~$\lambda (G)$ of a finite group~$G$ 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. The generalized Fitting height of a finite group~$G$ is the least number $h=h^*(G)$ such that $F^*_h(G)=G$, where $F^*_1(G)=F^*(G)$ is the generalized Fitting subgroup, and $F^*_{i+1}(G)$ is the inverse image of $F^*(G/F^*_{i}(G))$. 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~$A$.