Alphabeta Math
CounterexampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16
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 of an abelian group need not be characteristic

Statement refuted

Every subgroup of an abelian group is characteristic.

Facts & Assumptions

Given: The additive group V=C2×C2 and its subgroup K=⟨(1,0)⟩.

[L1]

A subgroup is characteristic when every automorphism of the ambient group maps it to itself (Characteristic subgroups).

[L2]

An automorphism is a bijective homomorphism from a group to itself (Group isomorphisms, automorphisms and the set Aut⁡(G)).

Counterexample

technique · direct
1.1

The coordinate swap u(x,y)=(y,x) preserves addition and is its own inverse, so it is an automorphism by [L2].

L2algebra
2.1

But u(K)=⟨(0,1)⟩≠K. Hence K is not characteristic by [L1], even though V is abelian and therefore every subgroup of V is normal.

step 1.1L1∎

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: 12 results over 9 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