 Order of a direct-product element

If the order of a group element is $m$ for $g$ and $n$ for $h$, with both finite, thenexactly when both $m$ and $n$ divide $k$. The smallest positive such $k$ is their least common multiple, givingThis concerns the orders of the selected elements, which need not equal the orders of their entire groups. If either component has infinite order of a group element, no positive power can return the pair to the identity element, so the pair also has infinite order.

For example, in $C_4\times C_6$, the additive element $(2,3)$ has component orders $2,2$, hence pair order $2$. The element $(1,1)$ instead has order $\mathrm{lcm}(4,6)=12$. The product has $24$ elements, so even this latter element does not generate it. This distinguishes group order from order of a group element while exposing the simultaneous-return interpretation.