Let be a group with . Assume and . Suppose that and commute; i.e., that . Prove that divides the least common multiple of and . Need this be true if and do not commute? Give an example of commuting elements and such that the order of is not equal to the least common multiple of and .
Supposing that , we saw in a previous theorem that for all . In particular, . Thus by a previous exercise, divides .
Consider . Note that , so that . However has order 3, and 3 does not divide 2.
For a trivial example, consider and let . Less trivially, consider , where and . Note that , while . Thus .