Sign of a permutation

The sign homomorphism $\mathrm{sgn}:S_n\to \{\pm 1\}$ is the determinant of the corresponding permutation matrix. A transposition has sign $-1$, so this also proves that the parity of any transposition decomposition is well-defined.

An odd permutation has sign $-1$, equivalently it is a product of an odd number of transpositions.

An even permutation has sign $+1$, equivalently it is a product of an even number of transpositions.