On the Order of Products of Coprime Elements in Finite Groups

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.