On the length of finite factorized groups

TL;DR

This paper establishes bounds on the nonsoluble length and generalized Fitting height of finite groups based on their factorization properties, specifically when factored by subgroups of coprime orders or soluble subgroups.

Contribution

It proves new bounds relating nonsoluble length and generalized Fitting height to subgroup properties in finite factored groups.

Findings

Nonsoluble length is bounded by generalized Fitting heights of factors.

Generalized Fitting height is bounded when one factor is soluble with known derived length.

Results apply to groups factorized by coprime order subgroups.

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 $h=h^*(G)$ 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 $A$.