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.

Divisibility of an image with torsion kernel

Statement

Let $f : G \to H$ be a homomorphism of abelian groups. If multiplication by every positive integer is bijective on $G$ and every element of $\ker f$ is torsion, then multiplication by every positive integer is bijective on the subgroup $im f \subseteq H$ .

Facts & Assumptions

Given: The map and hypotheses in the statement.

Goal: Prove unique divisibility of its image.

[F1]

A torsion kernel in a uniquely divisible group vanishes makes $f$ injective.

[F2]

An injection identifies its domain with its image identifies an injective map's domain with its image.

[F3]

Unique divisibility is invariant under group isomorphism transfers unique divisibility across an isomorphism.

Proof

1.1F1F2given

By [F1], the kernel hypothesis and unique divisibility make $f$ injective. Its corestriction to its image is therefore an isomorphism by [F2].

2.1F3step 1.1given∎

Apply [F3] to that isomorphism and the given multiplication maps on $G$ . Every positive multiplication map on $im f$ is bijective.

Depends on

Used by · 1

Unique divisibility of the smooth-point subgroup

Dependency tree · 2 levels

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