ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-11

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.

Amalgamating C_2 inside C_4 and C_6 gives the presentation with a^2=b^3

Example

Embed $C_{2}$ as the unique order-two subgroup of $C_{4} = ⟨ a ⟩$ and $C_{6} = ⟨ b ⟩$ . Their amalgamated free product has presentation $⟨ a, b ∣ a^{4} = e, b^{6} = e, a^{2} = b^{3} ⟩ .$

Facts & Assumptions

Given: The objects and hypotheses in the example.

[L1]

Let $G = ⟨ X ∣ R ⟩$ and $H = ⟨ Y ∣ S ⟩$ with disjoint generators, and let $f, h$ embed $K$ . If $T$ generates $K$ and words $u_{t} (X), v_{t} (Y)$ represent $f (t), h (t)$ , then $G *_{K} H ≅ ⟨ X ⊔ Y ∣ R \cup S \cup {u_{t} v_{t}^{- 1} : t \in T} ⟩ .$ (A free product with amalgamation has the factor presentations plus the amalgamating relations).

[L2]

The canonical maps $G \to G *_{K} H$ and $H \to G *_{K} H$ are injective. (The factor maps into a free product with amalgamation are injective).

[L3]

Inside $G *_{K} H$ , the images of $G$ and $H$ intersect exactly in their common image of $K$ . (The two factor images intersect exactly in the amalgamated subgroup).

[L4]

Every subgroup $H$ of a cyclic group $G = ⟨ g ⟩$ is cyclic. If $H \neq {e}$ , then the least positive integer $d$ for which $g^{d} \in H$ satisfies $H = ⟨ g^{d} ⟩$ . (Every subgroup of a cyclic group is cyclic; the least positive exponent in a nontrivial subgroup supplies a generator).

Verification

technique · direct

1.1

Enumerating the cyclic powers shows that $a^{2}$ is the unique element of order $2$ in $C_{4}$ and $b^{3}$ is the unique element of order $2$ in $C_{6}$ . The edge maps send the nonidentity element of $C_{2}$ to these elements, so both maps are injective.

givenL1L2L3L4

2.1

The amalgamated-presentation theorem gives the displayed presentation.

step 1.1

3.1

Factor embedding keeps copies of $C_{4}$ and $C_{6}$ , and the intersection theorem says their images meet exactly in the common $C_{2}$ .

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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.