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.

Every canonical factor map into a free product is injective

Statement

Every canonical factor homomorphism $ι_{i} : G_{i} \to *_{j} G_{j}$ is injective.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[L1]

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. (Normal form theorem for free products).

[L2]

A group homomorphism is injective if and only if its kernel is trivial. For a group homomorphism $f : G \to H$ , $f$ is injective exactly when $\ker f = {e_{G}}$ . (A group homomorphism is injective if and only if its kernel is trivial).

Proof

technique · direct

1.1

A nonidentity $g \in G_{i}$ maps to the nonempty one-syllable reduced word $(i, g)$ , which is nonidentity by normal form.

givenL1L2

2.1

Thus the kernel of $ι_{i}$ is trivial, and the trivial-kernel criterion gives injectivity. This also covers a trivial factor.

step 1.1∎

Depends on

Used by

Each factor is a retract of a free product when all other factors are sent trivially Corollary

Dependency tree · two levels

9 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)