The order of the product of two elements in finite nilpotent groups

TL;DR

This paper investigates how the order of a product of two elements in finite nilpotent groups relates to their mutual order, generalizing classical bounds from abelian groups to non-commutative cases and providing explicit descriptions for groups of class 2.

Contribution

It introduces the notion of mutual order in non-abelian groups and establishes bounds on the ratio of element orders in finite nilpotent groups of various classes.

Findings

The ratio of orders lies in a finite set depending on the group's class.

In p-groups with p > class, the orders are equal.

Explicit descriptions are provided for groups of class 2.

Abstract

An old problem in group theory is that of describing how the order of an element behaves under multiplication. To generalize some classical bounds concerning the order $\mathrm o(ab)$ of two elements $a, b$ in a finite abelian group to the non-commutative case, we replace $\mathrm o(ab)$ with a notion of mutual order $\mathrm o(a, b)$, defined as the least positive integer $n$ such that $a^nb^n = 1$. Motivated by this, we then compare $\mathrm o(ab)$ and $\mathrm o(a, b)$ in finite nilpotent groups, and show that in a group of class $\gamma$, the ratio $\mathrm o(ab)/\mathrm o(a, b)$ lies in some fixed finite set $S(\gamma) \subset \mathbb Q$, whose elements do not involve prime factors exceeding $\gamma$. In particular, we generalize a result of P. Hall, which asserts that $\mathrm o(ab) = \mathrm o(a, b)$ in $p$-groups with $p > \gamma$. We end with a more detailed analysis for groups of class 2, which allows one to give a more explicit description of $\mathrm o(ab)/\mathrm o(a, b)$.