Even and odd permutations

Definition

Let $n \geq 2$ and let $\sigma \in S_n$. The permutation $\sigma$ is even when $\operatorname{sgn}(\sigma) = 1$, and odd when $\operatorname{sgn}(\sigma) = -1$.

Remarks

The identity is even and each transposition is odd. A product of two even or two odd permutations is even, and a product of an even and an odd permutation is odd. For $n \geq 2$, $S_n$ has $n!/2$ even and $n!/2$ odd permutations. The odd permutations do not form a subgroup, but they form a coset of $A_n$.

Explanations

L1 human

Each permutation is even or odd, like each integer. Even and odd combine by the same rules as for integers. Exactly half of the permutations are even.