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CorollaryStatement: AI-adaptedProof: AI-generatedSession-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.

Every subgroup of an abelian group is normal

Statement

Every subgroup of an abelian group is normal.

Facts & Assumptions

Given: An abelian group GG and a subgroup H≤GH\le G.

[F1]

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

[L1]

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

Proof

technique · direct
1.1

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

F1algebra
2.1

Hence H⊴GH\mathrel{\trianglelefteq}G by the coset characterisation of normality.

step 1.1L1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 14 results over 13 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