LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 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.

${Core}_{G} (H)$ is the largest normal subgroup of $G$ contained in $H$

Statement

For $H \leq G$ , the core $K = {Core}_{G} (H)$ is a normal subgroup of $G$ , satisfies $K \leq H$ , and contains every normal subgroup of $G$ that is contained in $H$ . Thus it is the largest normal subgroup of $G$ contained in $H$ .

Facts & Assumptions

Given: A group $G$ , a subgroup $H \leq G$ , and $K : = {Core}_{G} (H)$ .

[L1]

The core is $K = ⋂_{g \in G} g H g^{- 1}$ (The core ${Core}_{G} (H) = ⋂_{g \in G} g H g^{- 1}$ of a subgroup).

[L2]

A subgroup $N$ is normal when $g N g^{- 1} = N$ for every $g \in G$ (Normal subgroup: invariance under conjugation).

[L3]

Normality is equivalent to $g N g^{- 1} \subseteq N$ for every $g \in G$ (Equivalent characterisations of a normal subgroup by conjugates and left and right cosets).

[L4]

A nonempty subset $S$ of a group is a subgroup if $a b^{- 1} \in S$ for every $a, b \in S$ (One-step subgroup test: a nonempty $H \subseteq G$ is a subgroup iff $g h^{- 1} \in H$ for all $g, h \in H$ ; the identity and the inverses of $H$ are then those of $G$ ).

Proof

technique · direct

1.1

The identity belongs to every $g H g^{- 1}$ , and if $a, b$ belong to every such conjugate then $a b^{- 1}$ does too; [L4] makes $K$ a subgroup. The factor for $g = e$ is $H$ , so $K \leq H$ .

L1L4

1.2

For $t \in G$ , conjugation sends the family ${g H g^{- 1} : g \in G}$ to ${(t g) H (t g)^{- 1} : g \in G}$ , the same family because $g \mapsto t g$ is a bijection; hence $t K t^{- 1} = K$ , and [L2] gives $K ⊴ G$ .

L1L2L3

2.1

If $N ⊴ G$ and $N \leq H$ , then $N = g N g^{- 1} \leq g H g^{- 1}$ for every $g \in G$ , so $N \leq ⋂_{g} g H g^{- 1} = K$ .

L1L2L3∎

Depends on

Used by

Cited to discharge well-definedness by The core Core_G(H)=⋂_g∈ GgHg⁻¹ of a subgroup.

Dependency tree · two levels

11 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

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

Core⁡G(H)CoreG​(H) is the largest normal subgroup of GG contained in HH