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
2026-04-02 00:34:54.1775090094
A conceptual problem in group theory
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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.