# Odd permutation

From Groupprops

## Definition

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

- It can be expressed as a product of an odd number of transpositions.
- The number of cycles of even length in its cycle decomposition is odd.
- 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).