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

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 $\mathrm o(ab)/\mathrm o(a, b)$.