Cycle Decomposition of Permutations is Unique

Cycle decomposition of Permutations into disjoint cycles is unique (up to reordering of cycles).

A proof can be found here.

The condition of disjoint is crucial. For example, the permutation (1 3 2) can be factored into (2 3)(1 2), where the two cycles are not disjoint. (1 3 2)=(1 2)(1 3) is also another decomposition, the two cycles are also not disjoint.

Wolframalpha can calculate permutations, a useful tool to replace manual calculations. Take note though that Wolframalpha’s convention is multiplying permutations from left to right, while most books follow the convention of multiplying right to left.

Recommended Math Books from Amazon


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