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An old problem in group theory is that of describing how the order of anelement behaves under multiplication. To generalize some classical boundsconcerning the order $\mathrm o(ab)$ of two elements $a, b$ in a finite abeliangroup to the non-commutative case, we replace $\mathrm o(ab)$ with a notion ofmutual order $\mathrm o(a, b)$, defined as the least positive integer $n$ suchthat $a^nb^n = 1$. 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 finiteset $S(\gamma) \subset \mathbb Q$, whose elements do not involve prime factorsexceeding $\gamma$. In particular, we generalize a result of P. Hall, whichasserts 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 togive a more explicit description of $\mathrm o(ab)/\mathrm o(a, b)$.

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