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.

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The core $\operatorname{Core}_G(H)=\bigcap_{g\in G}gHg^{-1}$ of a subgroup

Definition

Let $H\le G$ be a subgroup (Subgroup). Its core in $G$ is

$\operatorname{Core}_G(H):=\bigcap_{g\in G}gHg^{-1}.$

Each $gHg^{-1}$ is a subgroup because conjugation is an automorphism (Conjugation $x\mapsto gxg^{-1}$ is an automorphism). The facts that the displayed intersection is normal in $G$ and is the largest normal subgroup of $G$ contained in $H$ are proved in $\operatorname{Core}_G(H)$ is the largest normal subgroup of $G$ contained in $H$ ↗.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 14 results over 11 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

K. Conrad, Group Actions, Theorem 6.8 (standard reference, not scraped)

The core Core⁡G(H)=⋂g∈GgHg−1\operatorname{Core}_G(H)=\bigcap_{g\in G}gHg^{-1}CoreG​(H)=⋂g∈G​gHg−1 of a subgroup