On the Generalized Fitting Height and Nonsoluble Length of the Mutually Permutable Products of Finite Groups

TL;DR

This paper investigates the properties of the generalized Fitting height and non-soluble length in finite groups, especially focusing on groups formed as mutually permutable products of subgroups, and establishes bounds and equalities for these measures.

Contribution

It provides new bounds and equalities relating the generalized Fitting height and non-$p$-soluble length of groups formed by mutually permutable subgroups, extending understanding of their structural properties.

Findings

For mutually permutable products, the generalized Fitting height of the whole group is at most one more than the maximum of the subgroups' heights.

The non-$p$-soluble length of the product equals the maximum of the subgroups' lengths.

Introduces and studies the non-Frattini length in this context.

Abstract

The generalized Fitting height $h^*(G)$ of a finite group $G$ is the least number $h$ such that $\mathrm{F}_h^* (G) = G$, where $\mathrm{F}_{(0)}^* (G) = 1$, and $\mathrm{F}_{(i+1)}^*(G)$ is the inverse image of the generalized Fitting subgroup $\mathrm{F}^*(G/\mathrm{F}^*_{(i)} (G))$. Let $p$ be a prime, $1=G_0\leq G_1\leq\dots\leq G_{2h+1}=G$ be the shortest normal series in which for $i$ odd the factor $G_{i+1}/G_i$ is $p$-soluble (possibly trivial), and for $i$ even the factor $G_{i+1}/G_i$ is a (non-empty) direct product of nonabelian simple groups. Then $h=\lambda_p(G)$ is called the non-$p$-soluble length of a group $G$. We proved that if a finite group $G$ is a mutually permutable product of of subgroups $A$ and $B$ then $\max\{h^*(A), h^*(B)\}\leq h^*(G)\leq \max\{h^*(A), h^*(B)\}+1$ and $\max\{\lambda_p(A), \lambda_p(B)\}= \lambda_p(G)$. Also we introduced and studied the non-Frattini length.