False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-04

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.

The additive group of Zp is cyclic as an abstract group

Statement

The additive group of $Z_{p}$ is cyclic as an abstract group.

Facts & Assumptions

Given: The additive group of $Z_{p}$ .

[L1]

The element $1$ topologically generates $Z_{p}$ , but the additive group is not abstractly cyclic (The additive topological group of Zp is topologically generated by 1, although it is not abstractly cyclic).

[L2]

The additive group of $Z_{p}$ is torsion-free (The additive group of Zp is torsion-free).

Refutation

technique · direct

1.1

By [L1], the underlying set of $Z_{p}$ is uncountable, whereas the subgroup generated by any one element is the countable image of $Z$ . Thus no element generates $Z_{p}$ as an abstract group.

L1given

2.1

In particular, step 1.1 rules out every infinite cyclic realization, while [L2] rules out every nontrivial finite cyclic realization. Hence the claim is false.

L2step 1.1algebra

3.1

Therefore $Z_{p}$ is topologically generated by one but not cyclic as an abstract group.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Jordan Bell, The profinite completion of the integers, the p-adic integers, and Prufer p-groups (standard reference, not scraped)