ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

$Aut (C_{8}) ≅ C_{2} \times C_{2}$

Example

The automorphism group of the cyclic group of order eight is

$Aut (C_{8}) ≅ C_{2} \times C_{2} .$

Facts & Assumptions

Given: The cyclic group $C_{8}$ .

[L1]

$Aut (C_{8}) ≅ (Z / 8)^{\times}$ ( $Aut (C_{n}) ≅ (Z / n Z)^{\times}$ ).

[L2]

The units modulo $8$ are the residue classes coprime to $8$ (For $n \geq 1$ , $[a]_{n}$ is a unit if and only if $\gcd (a, n) = 1$ ).

Verification

technique · direct

1.1L2algebra

By [L2], $(Z / 8)^{\times} = {1, 3, 5, 7}$ . Each nonidentity element squares to $1$ modulo $8$ : $3^{2} \equiv 5^{2} \equiv 7^{2} \equiv 1$ .

2.1step 1.1L1algebra∎

The map from $C_{2} \times C_{2}$ sending $(0, 0), (1, 0), (0, 1), (1, 1)$ to $1, 3, 5, 7$ , respectively, is bijective, and direct multiplication modulo $8$ verifies that it preserves the group operation. Thus $(Z / 8)^{\times} ≅ C_{2} \times C_{2}$ , and [L1] gives the result.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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.

Aut⁡(C8)≅C2×C2Aut(C8​)≅C2​×C2​