Alphabeta Math
CorollaryStatement: AI-adaptedProof: AI-generatedprecheck 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.

Every subgroup of an abelian group is normal

Statement

Every subgroup of an abelian group is normal.

Facts & Assumptions

Given: An abelian group G and a subgroup H≤G.

[F1]

A group is abelian when xy=yx for all of its elements (Group and abelian group).

[L1]

A subgroup H≤G is normal if and only if gH=Hg for every g∈G (Equivalent characterisations of a normal subgroup by conjugates and left and right cosets).

Proof

technique · direct
1.1F1algebra

For every g∈G, commutativity gives gH={gh:h∈H}={hg:h∈H}=Hg.

2.1step 1.1L1∎

Hence H⊴G by the coset characterisation of normality.

Depends on

Dependency tree · 2 levels

Sources