Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 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.

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≤GN\le G. Then N⊴GN\mathrel{\trianglelefteq}G exactly when there are a group HH and a homomorphism f:G→Hf:G\to H with ker⁡f=N\ker f=N.

Facts & Assumptions

Given: A subgroup N≤GN\le 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⊴GN\mathrel{\trianglelefteq}G, the canonical map π:G→G/N\pi:G\to G/N is a homomorphism with kernel NN (The canonical projection π:G→G/N\pi:G\to G/N, π(g)=gN\pi(g)=gN, is a surjective group homomorphism).

[L3]

The kernel of ff 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⁡fN=\ker f for a homomorphism, then NN is normal by [L1].

L1L2L3L4given
2.1

If NN is normal, [L2] supplies the quotient homomorphism π\pi and gives ker⁡π=N\ker\pi=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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 32 results over 14 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