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.

A torsion kernel in a uniquely divisible group vanishes

Statement

Let $f : G \to H$ be a homomorphism of abelian groups. If multiplication by each positive integer is bijective on $G$ , and every element of $\ker f$ is killed by some positive integer, then $f$ is injective.

Facts & Assumptions

Given: The homomorphism and hypotheses in the statement.

Goal: Prove injectivity.

[F1]

Unique divisibility excludes torsion proves that $G$ is torsion-free under the stated hypothesis.

Proof

1.1F1given

Every element of $\ker f$ is torsion by hypothesis, so [F1] makes it zero. Thus $\ker f = 0$ .

2.1step 1.1algebra∎

If $f (a) = f (b)$ , then $a - b \in \ker f = 0$ by step 1.1, hence $a = b$ . Therefore $f$ is injective.

Depends on

Unique divisibility excludes torsion

Used by · 1

Divisibility of an image with torsion kernel

Dependency tree · 1 level

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.