---
tag: 0V0E04DWX2
kind: statement
role: proposition
name: Sign of a cycle
msc: [20B30]
uses: [0SK90238BH, 0XK6FECMW5]
sources:
  - url: https://en.wikipedia.org/wiki/Parity_of_a_permutation
    locator: Properties
  - url: https://en.wikipedia.org/wiki/Cyclic_permutation
    locator: Basic properties
---

## Statement

Let $k \geq 1$ and let $\sigma$ be a $k$-cycle in $S_n$.
Then $\operatorname{sgn}(\sigma) = (-1)^{k-1}$.

## Remarks

A $k$-cycle is a product of $k - 1$ transpositions, which gives the formula.
Thus a cycle is an [[even permutation|0SQTMQTSRZ]] exactly when its length is odd.
With the [[disjoint cycle decomposition|0YWDAZPRPQ]], the sign of $\sigma \in S_n$ is $(-1)^{n - c}$, where $c$ is the number of cycles, with the fixed points counted as $1$-cycles.

## Explanations

### L1 human

A cycle of length $k$ is a chain of $k - 1$ swaps.
So a $3$-cycle is even and a transposition is odd.
The length and the parity of a cycle are opposite.