Even permutation
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.
- Given a total ordering on the underlying set, the number of unordered pairs of elements for which the permutation is order-reversing is even.