[summary: In a symmetric group, if we are applying a collection of permutations which are each disjoint cycles, we get the same result no matter the order in which we perform the cycles.]
Consider two cycles and in the Symmetric group , where all the are distinct.
Then it is the case that the following two elements of are equal:
- , which is obtained by first performing the permutation notated by and then by performing the permutation notated by
- , which is obtained by first performing the permutation notated by and then by performing the permutation notated by
Indeed, (taking to be ), while , so they agree on elements of . Similarly they agree on elements of ; and they both do not move anything which is not an or a . Hence they are the same permutation: they act in the same way on all elements of .
This reasoning generalises to more than two disjoint cycles, to show that disjoint cycles commute.