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

TL;DR

This paper investigates the relationship between the nonsoluble length and generalized Fitting height of finite groups, establishing bounds based on properties of their 2-generator subgroups, and proposes related conjectures.

Contribution

It proves that the nonsoluble length of a finite group is bounded by that of its 2-generator subgroups and provides partial results towards a conjecture on generalized Fitting height.

Findings

If all 2-generator subgroups have nonsoluble length ≤ k, then the group has nonsoluble length ≤ k.

Under certain conditions, the generalized Fitting height of the group is bounded by that of its 2-generator subgroups.

The paper proposes a conjecture relating the generalized Fitting height of the group to that 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 $h=h^*(G)$ 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 $F^*(G/F^*_{i}(G))$. In the present paper we prove that if $\lambda (J)\leq k$ for every 2-generator subgroup $J$ of $G$, then $\lambda(G)\leq k$. It is conjectured that if $h^*(J)\leq k$ for every 2-generator subgroup $J$, then $h^*(G)\leq k$. We prove that if $h^*(\langle x,x^g\rangle)\leq k$ for all $x,g\in G$ such that $\langle x,x^g\rangle$ is soluble, then $h^*(G)$ is $k$-bounded.