CorollaryStatement: Literature-sourcedProof: 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.

The two factor images intersect exactly in the amalgamated subgroup

Statement

Inside $G *_{K} H$ , the images of $G$ and $H$ intersect exactly in their common image of $K$ .

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[L1]

Every element of $G *_{K} H$ has a unique normal form $s_{1} \dots s_{n} k$ relative to fixed transversals. A normal word of positive length is nonidentity. The represented group and these conclusions are independent of the chosen transversals. (Normal form theorem for free products with amalgamation).

[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).

Proof

technique · direct

1.1

The image of $K$ lies in both factor images by the commuting pushout square.

givenL1L2

2.1

If the images of $g \in G$ and $h \in H$ are equal, then the normal form of $g h^{- 1}$ is trivial. Normal-form uniqueness forces both factor representatives to reduce to the same length-zero element of $K$ .

step 1.1

3.1

Thus the intersection is precisely the amalgamated subgroup. The argument includes trivial $K$ and a whole-factor inclusion.

step 2.1∎

Depends on

Used by

Amalgamating C₂ inside C₄ and C₆ gives the presentation with a²=b³ Example

Dependency tree · two levels

7 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

George D. Torres, Combinatorial Group Theory, §2 (standard reference, not scraped)
B. H. Neumann, Lectures on Topics in the Theory of Infinite Groups, Ch. 9 (standard reference, not scraped)