TheoremStatement: 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.

Normal form theorem for free products

Statement

Every element of $*_{i \in I} G_{i}$ has a unique reduced syllable expression. The identity is represented by the empty word, and no nonempty reduced word represents the identity.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[L1]

The reduced syllable words in $(G_{i})_{i \in I}$ form a group under concatenation followed by seam reduction. The one-syllable maps $G_{i} \to W$ make this group a free product of the family. (Reduced syllable words form the free product of a family of groups).

[L2]

For groups as in def-group, a syllable is a tagged pair $(i, g)$ with $i \in I$ and $g \in G_{i} ∖ {e_{i}}$ . A reduced syllable word is a finite list of syllables, indexed by a natural length as in def-natural-numbers, in which adjacent tags differ. The empty list is allowed. At a concatenation seam, adjacent syllables from the same factor are multiplied and an identity result is deleted; this elementary reduction is repeated until the seam is reduced. (Reduced syllable words in a family of groups).

[L3]

For a family $(G_{i})_{i \in I}$ , a free product is a group $F$ with homomorphisms $ι_{i} : G_{i} \to F$ in the sense of def-group-homomorphism, such that for every group $H$ and every family of homomorphisms $f_{i} : G_{i} \to H$ , there is a unique homomorphism $f : F \to H$ satisfying $f \circ ι_{i} = f_{i}$ for all $i$ . It is denoted $*_{i \in I} G_{i}$ . Injectivity of the maps $ι_{i}$ is not part of this definition. (The free product of an arbitrary family of groups).

Proof

technique · direct

1.1

Let $F$ be any free product and $W$ the reduced-word model. Their universal properties give factor-compatible homomorphisms $F \to W$ and $W \to F$ . Each composite agrees with the identity on every factor, so uniqueness in the universal property makes the maps inverse isomorphisms.

givenL1L2L3

2.1

In the model every element is literally one reduced word. Distinct reduced words act differently on the empty word, so they are distinct elements.

step 1.1

3.1

Consequently the empty word is the identity, every nonempty reduced word is nonidentity, and the reduced expression is unique.

step 2.1∎

Depends on

Used by

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)