# Even permutation

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## Definition

An even permutation is a permutation on a finite set (equivalently, a finitary permutation on a possibly infinite set) satisfying the following equivalent conditions:

- It can be expressed as a product of an even number of transpositions.
- The number of cycles of even length in its cycle decomposition is even.
- It is in the alternating group (respectively, the finitary alternating group) on the set.