The order of the product of two elements in periodic groups

M. Amiri, I. Lima

Published 2020-09-01Version 1

An old problem in group theory is that of describing how the order of an element behaves under multiplication. Let $G$ be a periodic group, and let $LCM(G)$ be the set of all $x\in G$ such that $o(x)$ is less than $exp(G)$ and $o(xz)$ divides the least common multiple of $o(x)$ and $o(z)$ for all $z$ in $G$. In this article, we prove that the subgroup generated by $LCM(G)$ is a nilpotent characteristic subgroup of $G$ whenever $G$ is a solvable group or $G$ is a finite group.