ExampleConstruction: Literature-sourcedVerification: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Small elementary and hyperelementary groups

Example

For distinct primes $p, q$ , $C_{p q} ≅ C_{q} \times C_{p}$ is $p$ -elementary. For odd $p$ , $D_{2 p} = C_{p} ⋊ C_{2}$ is $2$ -hyperelementary and is $2$ -elementary exactly when the action is trivial. Thus $S_{3} = C_{3} ⋊ C_{2}$ is $2$ -hyperelementary but not $2$ -elementary. Likewise, the nontrivial $C_{7} ⋊ C_{3}$ is $3$ -hyperelementary but not $3$ -elementary.

Facts & Assumptions

[F1]

The cited prerequisite is $p$ -elementary and $p$ -hyperelementary finite groups.

Verification

Given: the displayed semidirect products use their indicated conjugation actions.

1.1

In each case the first factor is cyclic of order prime to the displayed prime and the second is a $p$ -group.

F1given

2.1

Directness is equivalent to trivial conjugation; the reflections in $S_{3}$ invert $C_{3}$ , and the chosen action of $C_{3}$ on $C_{7}$ is nontrivial, establishing the two non-elementary assertions. ∎

step 1.1

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Wen-Wei Li, Yanqi Lake Lectures on Algebra I, Definition 14.1.1 (standard reference, not scraped)