Odd permutation

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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).