DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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 normalizer $N_{G} (H) = {g \in G : g H g^{- 1} = H}$ of a subgroup

Definition

Let $H \leq G$ be a subgroup (Subgroup). The normalizer of $H$ in $G$ is

$N_{G} (H) : = {g \in G : g H g^{- 1} = H} .$

Thus $g \in N_{G} (H)$ exactly when the conjugation automorphism $c_{g}$ preserves $H$ setwise (Conjugation $x \mapsto g x g^{- 1}$ is an automorphism). The subgroup property is proved in $C_{G} (x)$ and $N_{G} (H)$ are subgroups of $G$ ↗.

Depends on

Dependency tree · 2 levels

Sources

P. Brosnan, Undergraduate Algebra Notes, 3.14: G-Sets