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TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-03
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Free groups on the same set are uniquely isomorphic compatibly with their generators

Statement

If (F,i)(F,i) and (F′,i′)(F',i') are free groups on the same set XX, then there is a unique group isomorphism ϕ:F→F′\phi:F\to F' such that

ϕ∘i=i′.\phi\circ i=i'.

Facts & Assumptions

Given: Two free groups (F,i)(F,i) and (F′,i′)(F',i') on XX.

[L1]

A map from the generators of a free group extends uniquely to a group homomorphism (Free group on a set of generators).

[L2]

A group isomorphism is a bijective group homomorphism (Group isomorphisms, automorphisms and the set Aut⁡(G)\operatorname{Aut}(G)).

Proof

technique · constructive
1.1

Apply the universal property of FF to i′:X→F′i':X\to F' and construct the unique homomorphism ϕ:F→F′\phi:F\to F' with ϕi=i′\phi i=i'.

L1givenconstruct
1.2

Apply the universal property of F′F' to i:X→Fi:X\to F and construct the unique homomorphism ψ:F′→F\psi:F'\to F with ψi′=i\psi i'=i.

L1givenconstruct
2.1

Both ψϕ\psi\phi and id⁡F\operatorname{id}_F are homomorphisms F→FF\to F whose composites with ii equal ii, so uniqueness in the universal property gives ψϕ=id⁡F\psi\phi=\operatorname{id}_F.

step 1.1step 1.2L1
2.2

Symmetrically, ϕψ=id⁡F′\phi\psi=\operatorname{id}_{F'}.

step 1.1step 1.2L1
3.1

Thus ϕ\phi is bijective, hence a group isomorphism.

step 2.1step 2.2L2
4.1

Any generator-compatible homomorphism F→F′F\to F' equals ϕ\phi by the uniqueness in step 1.1; in particular the displayed isomorphism is unique.

step 1.1step 3.1L1L2discharge-construct: final∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 14 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