CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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A $k$ -cycle has sign $(-1)^{k-1}$ , and $\operatorname{sgn}(\sigma)=(-1)^{n-c(\sigma)}$ when fixed points are counted as cycles

Statement

A cycle of length $k$ has sign $(-1)^{k-1}$ . If $\sigma\in S_n$ and $c(\sigma)$ is the number of cycles after every fixed point is included as a one-cycle, then

$\operatorname{sgn}(\sigma)=(-1)^{n-c(\sigma)}.$

Facts & Assumptions

Given: A natural $n$ and a permutation $\sigma\in S_n$ .

[L1]

Sign is a homomorphism, every finite permutation has a disjoint-cycle decomposition, and a $k$ -cycle is a product of $k-1$ transpositions (The sign is a homomorphism $S_n\to\{+1,-1\}$ , surjective exactly when $n\ge 2$ , Every permutation of a finite set is a product of pairwise disjoint cycles, uniquely up to reordering and cyclic rotation, Every finite permutation is a product of transpositions, so the transpositions generate $S_n$ ).

Proof

technique · direct

1.1

The standard factorisation of a $k$ -cycle has $k-1$ transpositions, so [L1] gives sign $(-1)^{k-1}$ .

givenL1

2.1

Write the disjoint-cycle decomposition of $\sigma$ with lengths $k_1,\ldots,k_r$ . Multiplicativity of sign and step 1.1 give $\operatorname{sgn}(\sigma)=(-1)^{\sum_i(k_i-1)}$ .

step 1.1L1

3.1

Insert each fixed point as a one-cycle. Then the cycle lengths sum to $n$ , the number of cycles is $c(\sigma)$ , and $\sum_i(k_i-1)=n-c(\sigma)$ , which gives the formula.

step 2.1L1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 44 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. S. Milne, Group Theory, Corollary 4.27 (standard reference, not scraped)

A kkk-cycle has sign (−1)k−1(-1)^{k-1}(−1)k−1, and sgn⁡(σ)=(−1)n−c(σ)\operatorname{sgn}(\sigma)=(-1)^{n-c(\sigma)}sgn(σ)=(−1)n−c(σ) when fixed points are counted as cycles