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

Depends on

• Inversions, inversion number, the sign $\operatorname{sgn}(\sigma)=(-1)^{\operatorname{inv}(\sigma)}$, and even and odd permutations
• Every transposition factorisation of $\sigma$ has parity $(-1)^{\operatorname{inv}(\sigma)}$
• Every finite permutation is a product of transpositions, so the transpositions generate $S_n$
• Monoid homomorphism and group homomorphism
• The integers form a commutative ring

Used by

• A k-cycle has sign (-1)ᵏ⁻¹, and sgn(σ)=(-1)^n-c(σ) when fixed points are counted as cycles Corollary
• Aₙ is normal in Sₙ; for n≥2, 2 |Aₙ|=n!, while Aₙ=Sₙ for n=0,1 Corollary
• For n≥2, sign is the unique nontrivial homomorphism Sₙ→{+1,-1} Corollary
• For n≥5, the only proper nontrivial normal subgroup of Sₙ is Aₙ Corollary
• Cyclic sylow does not alone imply a normal p complement Counterexample
• Symmetrization and alternation operators Definition
• The alternating group Aₙ=ker(sgn) of even permutations Definition
• The sign representation of Sₙ and the restriction Res^G_H(V) of a representation to a subgroup Definition
• For n≥2, Sₙ≅ Aₙ⋊ C₂ using any transposition complement Example
• Every alternating multilinear F satisfies F(A)=F(I)∑_σ∈ Sₙsgn(σ)∏ᵢ a_σ(i),i Lemma
• Ordered and alternating Čech complexes agree Lemma
• Tail-swapping is a sign-reversing involution on the intersecting systems Lemma
• The symmetric groups Sₙ are solvable for n≤ 4 Lemma
• The Vandermonde product transforms by the sign of the root permutation Proposition
• Aₙ is generated by 3-cycles for every n≥3 Theorem
• det(lvertM(Aᵢ,Eⱼ)|)_i,j=∑_π∈ Sᵣsgn(π)·#{non-intersecting π-systems} Theorem
• Every group of order 30 has normal Sylow 3- and 5-subgroups and is not simple Theorem
• For every square matrix over a commutative ring, det(A^T)=det(A) Theorem
• For n≥2, an Sₙ-class of an even permutation splits in Aₙ exactly when all cycle lengths, including 1-cycles, are odd and distinct Theorem
• The Leibniz determinant is column-multilinear, alternating and normalized over every commutative ring Theorem

Dependency tree · two levels

27 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on 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)