TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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The sign is a homomorphism $S_n\to\{+1,-1\}$ , surjective exactly when $n\ge 2$

Statement

For every natural $n$ , the function $\operatorname{sgn}:S_n\to\{+1,-1\}$ is a group homomorphism. It is surjective exactly when $n\ge2$ ; for $n=0$ and $n=1$ its image is $\{1\}$ .

Facts & Assumptions

Given: A natural $n$ and permutations $\sigma,\rho\in S_n$ .

[L1]

Every finite permutation has a transposition factorisation, the sign is $(-1)^{\operatorname{inv}(\sigma)}$ , and every such factorisation has that parity (Every finite permutation is a product of transpositions, so the transpositions generate $S_n$ , Inversions, inversion number, the sign $\operatorname{sgn}(\sigma)=(-1)^{\operatorname{inv}(\sigma)}$ , and even and odd permutations, Every transposition factorisation of $\sigma$ has parity $(-1)^{\operatorname{inv}(\sigma)}$ ).

Proof

technique · direct

1.1

Choose transposition factorisations $\sigma=\tau_1\cdots\tau_r$ and $\rho=\upsilon_1\cdots\upsilon_s$ . Their concatenation is a transposition factorisation $\sigma\rho=\tau_1\cdots\tau_r\upsilon_1\cdots\upsilon_s$ in the library's composition order.

givenL1

2.1

By [L1], $\operatorname{sgn}(\sigma\rho)=(-1)^{r+s}=(-1)^r(-1)^s=\operatorname{sgn}(\sigma)\operatorname{sgn}(\rho)$ , and the identity has sign $1$ ; hence sign is a group homomorphism.

step 1.1L1

3.1

If $n\ge2$ , the transposition $(0\,1)$ belongs to $S_n$ and has sign $-1$ , while the identity has sign $1$ , so sign is surjective. If $n=0$ or $n=1$ , $S_n$ contains only the identity and the image is $\{1\}$ .

step 2.1L1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 56 results over 18 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, §4, the sign homomorphism (standard reference, not scraped)
Stanford Math 51H, Permutations (standard reference, not scraped)

The sign is a homomorphism Sn→{+1,−1}S_n\to\{+1,-1\}Sn​→{+1,−1}, surjective exactly when n≥2n\ge 2n≥2