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 excludes torsion

Statement

Let $G$ be an abelian group. Suppose that for every integer $n \geq 1$ , multiplication by $n$ is a bijection on $G$ . Then $G$ is torsion-free: if $n g = 0$ for some $n \geq 1$ , then $g = 0$ .

Facts & Assumptions

Given: The abelian group and bijectivity hypotheses in the statement.

Goal: Prove the stated torsion-freeness.

Proof

1.1given

For any $n \geq 1$ , the multiplication map $n : G \to G$ is injective by the given bijectivity. It sends $0$ to $0$ .

2.1step 1.1∎

If $n g = 0$ , then $n g = n 0$ ; injectivity from step 1.1 gives $g = 0$ . This holds for every positive $n$ , proving the claim.

Used by · 1

A torsion kernel in a uniquely divisible group vanishes