TITLE: A conceptual problem in group theory
QUESTION [8 upvotes]: As we all know that in group $S_n$ every pair of distinct  disjoint cycles commute .my doubt is is it reverse all true,mean if a pair of distinct cycles commute ,then they have to be disjoint??.i tried to find examples where distinct cycles commute but not disjoint,but fail to do so

REPLY [2 votes]: As suggested in the comments, the answer is that cycles $\sigma$ and $\tau$ commute if and only if either:
(i) $\sigma$ and $\tau$ are disjoint; or
(ii) $\sigma$ is a power of $\tau$ and $\tau$ is a power of $\sigma$.
In case (ii), this implies that they have the same order and hence the same length, and they must both be cycles on the same set of points.