Disjoint cy­cles com­mute in sym­met­ric groups

WikiLast edit: 14 Jun 2016 16:53 UTC by Patrick Stevens

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:

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.

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