Sign of a cycle

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 exactly when its length is odd. With the disjoint cycle decomposition, 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.