ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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.

Blocks in a regular cyclic action are cosets of subgroups

Example

Let the cyclic group $C_{n} = Z / n Z$ act on itself by translations $a \cdot x : = a + x .$ This action is regular (Regular actions).

Its blocks are exactly the cosets of subgroups of $C_{n}$ . In particular, if $n$ is composite, the cosets of any proper nontrivial subgroup give a nontrivial block system, while if $n$ is prime only the trivial block systems occur.

Facts & Assumptions

Given: The translation action of $C_{n}$ on itself.

[L1]

A block is a nonempty subset $B$ such that for every group element $g$ , either $g \cdot B = B$ or $(g \cdot B) \cap B = \emptyset$ (Blocks and block systems for a group action).

[L2]

A regular action is transitive and free (Regular actions).

Verification

technique · direct

1.1

If $H \leq C_{n}$ , then every translate $a + H$ is either $H$ itself or a disjoint coset of $H$ . So each coset of $H$ is a block, and the cosets of $H$ form a block system.

L1algebra

1.2

Conversely, let $B$ be a block containing $0$ . For any $x \in B$ , the translate $x + B$ meets $B$ at $x$ , so [L1] gives $x + B = B$ . Hence $B$ is closed under subtraction: if $x, y \in B$ , then $y \in x + B$ implies $y - x \in B$ . Therefore $B$ is a subgroup of $C_{n}$ .

2.1

Every block is a translate of one containing $0$ , so steps 1.1-1.2 show that the blocks are exactly the cosets of subgroups. When $n$ is composite, $C_{n}$ has a proper nontrivial subgroup; when $n$ is prime, it does not.

step 1.1step 1.2L2∎

Depends on

Used by

FALSE: every transitive action is primitive False statement
FALSE: transitivity alone forces nontrivial normal subgroups to be transitive False statement

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

J. S. Milne, Group Theory, Chapter 4 (standard reference, not scraped)
K. Conrad, Transitive Group Actions (standard reference, not scraped)