DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 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.

The alternating group $A_n=\ker(\operatorname{sgn})$ of even permutations

Definition

For $n\in\mathbb N$ , the alternating group is the kernel of the sign homomorphism,

$A_n:=\ker(\operatorname{sgn}:S_n\to\{+1,-1\})=\{\sigma\in S_n:\operatorname{sgn}(\sigma)=1\}.$

Thus $A_n$ consists exactly of the even permutations. The subgroup and normality assertions implicit in the word “group” follow from The image of a group homomorphism is a subgroup and its kernel is a normal subgroup.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 45 results over 13 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

T. W. Judson, Abstract Algebra: Theory and Applications, §5.1 (standard reference, not scraped)

The alternating group An=ker⁡(sgn⁡)A_n=\ker(\operatorname{sgn})An​=ker(sgn) of even permutations