Odd permutation

Definition
An odd permutation is a permutation on a finite set (equivalently, a finitary permutation on a set) satisfying the following equivalent conditions:


 * 1) It can be expressed as a product of an odd number of transpositions.
 * 2) The number of cycles of even length in its cycle decomposition is odd.
 * 3) It is in the symmetric group but not in the alternating group (equivalently, it is in the finitary symmetric group but not in the finitary alternating group).