On the Order of Products of Coprime Elements in Finite Groups

TL;DR

This paper introduces new subgroup structures in finite groups based on coprime element products, characterizes Frobenius decompositions, and extends the analysis to solvable groups with bounded Fitting height.

Contribution

It defines the subgroups $D_m(G)$ and $D_{m,n}(G)$, proves their properties, and introduces the $E$-series to classify certain solvable groups.

Findings

$D_m(G)$ and $D_{m,n}(G)$ are characteristic subgroups.

$D_{m,n}(G)$ is always nilpotent.

Groups with an $E$-series of length at most 4 have a specific characteristic subgroup structure.

Abstract

In this work, we introduce the subgroups $D_m(G)$ and $D_{m,n}(G)$, defined in terms of the orders of products of coprime elements in a finite group $G$. We show that both subgroups are characteristic, that $D_{m,n}(G)$ is always nilpotent, and that their nilpotent structure provides a characterization of Frobenius group decompositions. Furthermore, we define the $E$-series, which extends this framework to the study of an important class of solvable groups of Fitting height at most $4$. We prove that a finite group $G$ has an $E$-series of length at most $4$ if and only if there exists a characteristic subgroup $F \leq G$ such that $G/F$ is nilpotent and $F$ is either nilpotent, a Frobenius group, or a $2$-Frobenius group.