CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

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For $n \geq 2$ , sign is the unique nontrivial homomorphism $S_{n} \to {+ 1, - 1}$

Statement

For $n \geq 2$ , the sign homomorphism is the unique nontrivial group homomorphism $S_{n} \to {+ 1, - 1}$ .

Facts & Assumptions

Given: A natural $n \geq 2$ and a group homomorphism $φ : S_{n} \to {+ 1, - 1}$ .

[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 $n \geq 2$ ).

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 $π$ with $π (a b) π^{- 1} = (c d)$ .

givenL1

2.1

Since ${+ 1, - 1}$ is abelian, $φ (π τ π^{- 1}) = φ (τ)$ ; hence $φ$ takes one common value on every transposition.

step 1.1L1

3.1

By [L1], if the common value is $1$ then $φ$ 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 $φ = sgn$ , and sign is the unique nontrivial homomorphism.

step 2.1L1∎

Depends on

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Sources

J. S. Milne, Group Theory, Remark 4.25 (standard reference, not scraped)

For n≥2n≥2, sign is the unique nontrivial homomorphism Sn→{+1,−1}Sn​→{+1,−1}