LemmaStatement: AI-generatedProof: AI-generated

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.

Unique divisibility is invariant under group isomorphism

Statement

Let $u : G \to H$ be an isomorphism of abelian groups. If multiplication by every positive integer is bijective on $G$ , then the same is true on $H$ .

Facts & Assumptions

Given: The isomorphism and bijectivity hypotheses in the statement.

Goal: Transfer bijectivity to $H$ .

Proof

1.1givenalgebra

For each integer $n \geq 1$ , homomorphism compatibility gives $n_{H} = u \circ n_{G} \circ u^{- 1}$ .

2.1step 1.1∎

Each factor on the right is bijective, so $n_{H}$ is bijective. This holds for every positive integer, as required.

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