DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-02

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.

Normal subgroup: invariance under conjugation

Let $G$ be a group and let $N \leq G$ be a subgroup (Subgroup). For $g \in G$ , write

$g N g^{- 1} : = {g n g^{- 1} : n \in N} .$

The subgroup $N$ is normal in $G$ when

$g N g^{- 1} = N for every g \in G .$

In that case write $N ⊴ G$ . Equivalently, every inner conjugation of $G$ maps $N$ onto itself. The connection with equality of the left and right cosets of Left and right cosets $g H$ and $H g$ of a subgroup is proved in Equivalent characterisations of a normal subgroup by conjugates and left and right cosets.

Depends on