CorollaryStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Every canonical factor map into a free product is injective

Statement

Every canonical factor homomorphism $\iota_i:G_i\to\ast_jG_j$ is injective.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[L1]

Every element of $\ast_{i\in I}G_i$ 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 $\iota_i$ 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 29 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)