ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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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.

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The free group on the empty set is the trivial group

Example

Every free group on the empty set is trivial: it has exactly one element.

Facts & Assumptions

Given: The word-quotient model $F_{\mathrm{word}}(\varnothing)$ of The word-quotient group $W(X)/{\sim}$ satisfies the universal property of the free group on $X$ .

[L1]

Every class in $W(X)/{\sim}$ contains exactly one reduced word (Every class in $W(X)/{\sim}$ contains exactly one reduced word).

[L2]

If $(F,i)$ and $(F',i')$ are free groups on the same set $X$ , then there is a unique group isomorphism $\phi:F\to F'$ with $\phi\circ i=i'$ (Free groups on the same set are uniquely isomorphic compatibly with their generators).

[L3]

The word-quotient group $W(X)/{\sim}$ , with $x\mapsto[x]$ , is a free group on $X$ (The word-quotient group $W(X)/{\sim}$ satisfies the universal property of the free group on $X$ ).

Verification

technique · direct

1.1

On the empty alphabet, the empty word is the only finite word and hence the only reduced word; by [L1], $F_{\mathrm{word}}(\varnothing)$ has the single class $[\varepsilon]$ .

L1given

2.1

That class is the identity, so $F_{\mathrm{word}}(\varnothing)$ has exactly one element.

step 1.1given

3.1

By [L3] this model is a free group on $\varnothing$ , so [L2] makes every free group on $\varnothing$ isomorphic to it; an isomorphism is a bijection, so every free group on $\varnothing$ has exactly one element and is trivial.

L2L3step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 24 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

Encyclopedia of Mathematics, Free group (standard reference, not scraped)