On the Order of Products of Coprime Elements in Finite Groups

TL;DR

This paper addresses how the orders of products of coprime elements constrain the structure of finite groups by introducing the subgroups $D_m(G)$ and $D_{m,n}(G)$ and studying their properties. It shows these subgroups are characteristic, with $D_{m,n}(G)$ always nilpotent when $n>1$, and connects their nilpotent structure to Frobenius-type decompositions, notably as kernels in Frobenius groups. The authors then define the $E$-series, alternating the operators $D_{m,n}(\cdot)$ and $D_{n,m}(\cdot)$ to extract successive nilpotent layers, and prove that any finite group with such a series has Fitting height at most $4$. They provide a complete classification: $E$-series length $2$ corresponds to nilpotent groups, length $3$ to Frobenius groups, and length $4$ to $2$-Frobenius groups. This framework unifies element-order considerations with Frobenius kernels and classical Fitting theory, offering a structural lens for analyzing solvable finite groups.

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.

Theorems & Definitions (47)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Lemma 2.2
• proof
• Example 2.3
• Remark 2.4
• Lemma 2.5