ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck 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 $F_{w o r d} (\emptyset)$ 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 $ϕ : F \to F^{'}$ with $ϕ \circ i = i^{'}$ (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], $F_{w o r d} (\emptyset)$ has the single class $[ε]$ .

L1given

2.1

That class is the identity, so $F_{w o r d} (\emptyset)$ has exactly one element.

step 1.1given

3.1

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

L2L3step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

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