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

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Free products are unique up to a unique factor-compatible isomorphism

Statement

Any two free products of the same family are connected by a unique isomorphism commuting with every canonical factor map.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[L1]

For a family $(G_i)_{i\in I}$ , a free product is a group $F$ with homomorphisms $\iota_i:G_i\to F$ 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 $f\circ\iota_i=f_i$ for all $i$ . It is denoted $\ast_{i\in I}G_i$ . Injectivity of the maps $\iota_i$ is not part of this definition. (The free product of an arbitrary family of groups).

[L2]

Group isomorphisms, automorphisms and the set $\operatorname{Aut}(G)$ . An isomorphism $f:G\to H$ is a bijective group homomorphism (def-group-homomorphism, def-injection-surjection-bijection). When $G=H$ , it is an automorphism of $G$ . Write $\operatorname{Aut}(G):=\{f:G\to G:f\text{ is an automorphism}\}.$ (Group isomorphisms, automorphisms and the set $\operatorname{Aut}(G)$ ).

Proof

technique · direct

1.1

The universal properties give unique factor-compatible homomorphisms $u:F\to F'$ and $v:F'\to F$ .

givenL1L2

2.1

Both $v\circ u$ and $\mathrm{id}_F$ agree with every factor map, so uniqueness gives $v\circ u=\mathrm{id}_F$ ; similarly $u\circ v=\mathrm{id}_{F'}$ .

step 1.1

3.1

Hence $u$ is the unique compatible isomorphism. For the empty family both free products are trivial.

step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 10 results over 7 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)