Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-29
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.

A Kurosh decomposition of a subgroup of a free product

Example

Let G=C2∗C3 and let H=C2 be the first factor. Then the Kurosh decomposition of H is just H itself, with trivial free part.

Facts & Assumptions

Given: The Kurosh subgroup theorem.

[L1]

A subgroup of a free product is a free product of intersections with conjugates of the factors together with a free group. (Kurosh subgroup theorem)

Verification

technique · direct
1.1L1given

The subgroup H=C2 already is one of the free-product factors of G=C2∗C3. Therefore one of the Kurosh intersection terms in [L1] is exactly H.

2.1step 1.1algebra∎

No further nontrivial intersection factor is needed, and there is no free remainder. So the Kurosh decomposition here is the tautological one H≅C2.

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