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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

For $n\ge2$ , sign is the unique nontrivial homomorphism $S_n\to\{+1,-1\}$

Statement

For $n\ge2$ , the sign homomorphism is the unique nontrivial group homomorphism $S_n\to\{+1,-1\}$ .

Facts & Assumptions

Given: A natural $n\ge2$ and a group homomorphism $\varphi:S_n\to\{+1,-1\}$ .

[L1]

The transpositions generate $S_n$ , and sign is a homomorphism sending every transposition to $-1$ (Every finite permutation is a product of transpositions, so the transpositions generate $S_n$ , The sign is a homomorphism $S_n\to\{+1,-1\}$ , surjective exactly when $n\ge 2$ ).

Proof

technique · direct

1.1

Any two transpositions are conjugate in $S_n$ : for their two-point supports, the identity handles equality, a transposition handles one common point, and the product of two disjoint transpositions handles disjoint supports, producing a permutation $\pi$ with $\pi(a\,b)\pi^{-1}=(c\,d)$ .

givenL1

2.1

Since $\{+1,-1\}$ is abelian, $\varphi(\pi\tau\pi^{-1})=\varphi(\tau)$ ; hence $\varphi$ takes one common value on every transposition.

step 1.1L1

3.1

By [L1], if the common value is $1$ then $\varphi$ is trivial. If it is $-1$ , a product of $r$ transpositions has image $(-1)^r$ , which is also its image under sign because sign sends every transposition to $-1$ . Thus $\varphi=\operatorname{sgn}$ , and sign is the unique nontrivial homomorphism.

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 58 results over 20 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, Remark 4.25 (standard reference, not scraped)

For n≥2n\ge2n≥2, sign is the unique nontrivial homomorphism Sn→{+1,−1}S_n\to\{+1,-1\}Sn​→{+1,−1}