ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck 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 $c_{g} (x) = g x g^{- 1}$ (Inner automorphisms and $Inn (G)$ ).

[L2]

In an abelian group $g x = x g$ 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 $g x g^{- 1} = x g g^{- 1} = x$ .

L1L2L3givenalgebra

2.1

Hence $c_{g} = {id}_{G}$ 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 · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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