False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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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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False: an abelian group must have an abelian automorphism group

Statement

False claim: if a group $A$ is abelian, then $Aut (A)$ is abelian.

Facts & Assumptions

Given: The abelian group $V = C_{2} \times C_{2}$ .

[A1]

The false claim says that $Aut (V)$ is abelian.

[L1]

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

[L2]

The symmetric group on a set consists of all its permutations under composition (The symmetric group $Sym (X)$ : the bijections of a set $X$ under composition).

Refutation

technique · direct

1.1

Every automorphism of $V$ fixes the identity and permutes the three nonidentity elements. Conversely, every permutation of those elements preserves the group law because the sum of two distinct nonidentity elements is the third. Thus restriction gives $Aut (V) ≅ S_{3}$ by [L1] and [L2].

L1L2algebra

2.1

In $S_{3}$ , the transpositions $(12)$ and $(23)$ do not commute. Hence $Aut (V)$ is nonabelian, contradicting [A1] and refuting the claim.

step 1.1A1L2algebra∎

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: 11 results over 8 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

J. S. Milne, Group Theory (standard reference, not scraped)