On the length of a finite group and of its 2-generator subgroups

Abstract

The nonsoluble length $\lambda (G)$ of a finite group $G$ is defined as the minimum number of nonsoluble factors in a normal series of $G$ each of whose quotients 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 $F^{\ast }(G/F_i^{\ast }(G))$. In the present paper we prove that if $\lambda (J)\le k$ for every 2-generator subgroup $J$ of $G$, then $\lambda (G)\le k$. It is conjectured that if $h^{\ast }(J)\le k$ for every 2-generator subgroup $J$, then $h^{\ast }(G)\le k$. We prove that if $h^{\ast }(⟨x,x^g⟩)\le k$ for all $x,g\in G$ such that $⟨x,x^g⟩$ is soluble, then $h^{\ast }(G)$ is $k$-bounded.