> [!NOTE] Definition (Parity of Permutations) > Let $S_{n}$ denote the [[Symmetric Groups of Finite Degree|nth symmetric group]]. Let $\rho\in S_{n}$ be a [[Permutation of Finite Degree|permutation of n letters]]. Then $\rho$ is even iff it can be expressed as a product of an even number of [[Cyclic Permutation of n Letters|2-cycles]] and it is odd iff it can be expressed as a product of an odd number of $2$-cycles. **Note**: The [[Sign of Permutation of n Letters|sign]] of $\rho,$ denoted $\text{sgn}(\rho),$ is defined as $1$ iff $\rho$ is even and $-1$ iff $\rho$ is odd. # Properties **Well-defined**: [[Parity of Permutation on n Letters is Well-Defined]]. Note [[Parity of the inverse of a permutation]].