CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-02

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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.

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A subgroup is normal if and only if it is the kernel of a group homomorphism

Statement

A subgroup is normal if and only if it is the kernel of a group homomorphism.

Let $N \leq G$ . Then $N ⊴ G$ exactly when there are a group $H$ and a homomorphism $f : G \to H$ with $\ker f = N$ .

Facts & Assumptions

Given: A subgroup $N \leq G$ .

[L1]

The kernel of every group homomorphism is normal (The image of a group homomorphism is a subgroup and its kernel is a normal subgroup).

[L2]

If $N ⊴ G$ , the canonical map $π : G \to G / N$ is a homomorphism with kernel $N$ (The canonical projection $π : G \to G / N$ , $π (g) = g N$ , is a surjective group homomorphism).

[L3]

The kernel of $f$ consists of the elements sent to the identity (The kernel and image of a group homomorphism).

[L4]

Normality means invariance under conjugation (Normal subgroup: invariance under conjugation).

Proof

technique · direct

1.1

If $N = \ker f$ for a homomorphism, then $N$ is normal by [L1].

L1L2L3L4given

2.1

If $N$ is normal, [L2] supplies the quotient homomorphism $π$ and gives $\ker π = N$ .

step 1.1L1L2L3L4given

3.1

Thus normal subgroups are exactly kernels.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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

Milne, Group Theory, Kernels and Quotients (standard reference, not scraped)