---
tag: 0771AJP4BX
kind: definition
name: Even and odd permutations
aliases: [parity-of-a-permutation]
msc: [20B30]
defines: 0SQTMQTSRZ
uses: [0SK90238BH]
sources:
  - url: https://en.wikipedia.org/wiki/Parity_of_a_permutation
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Parity_of_a_permutation
    locator: Properties
---

## 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|0FWVRHBRWM]] 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.