Suppose $a$ and $b$ are group elements, $a$ and $b$ commute, $|a|$ and $|b|$ are both finite. What are the possibilities for $|ab|$?

This is an exercise in Gallian "Contemporary Abstract Algebra". Here $|a|$ and $|b|$ means the order of $a$ and the order of $b$. I know the order of a group element is the smallest positive integer n such that $a^n=e$ where $e$ is the identity element of the group.

I know that since $a$ and $b$ commute then $(ab)^n = a^nb^n$. I sketched a regular hexagon and labeled the vertices $0,1,2,\cdots,5$. I do not see any relationship between the orders of the elements and the order of their products in $C_6$. I stared at a list of the elements (and their orders) of the modulo $31$ multiplicative group. I don't see anything that helps me answer this question. Also, I should mention this exercise is in chapter $4$, Cyclic Groups, of the text and nothing very "deep" has been discussed in the previous chapters.

Solution 1:

You are very close. As you say $(ab)^n=a^nb^n=????$ That restricts the possibilities for $|ab|$ quite a bit. Can you show that all of them can be attained? In addition modulo $n$ all elements commute, which is handy.