ExampleConstruction: AI-adaptedVerification: 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.

Every inner automorphism of an abelian group is the identity

Example

Every inner automorphism of an abelian group is the identity.

Facts & Assumptions

Given: An abelian group $G$ and $g\in G$ .

[L1]

An inner automorphism has the form $c_g(x)=gxg^{-1}$ (Inner automorphisms and $\operatorname{Inn}(G)$ ).

[L2]

In an abelian group $gx=xg$ for all $g,x\in G$ (Group and abelian group).

[L3]

The center consists of elements commuting with every element (The center $Z(G)$ of a group).

Verification

technique · direct

1.1

Since $G$ is abelian, $g\in Z(G)$ and $gxg^{-1}=xgg^{-1}=x$ .

L1L2L3givenalgebra

2.1

Hence $c_g=\operatorname{id}_G$ for the chosen $g$ .

step 1.1L1L2L3given

3.1

As $g$ was arbitrary, every inner automorphism is the identity.

step 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: 18 results over 12 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

Milne, Group Theory, Automorphisms of Groups (standard reference, not scraped)