CorollaryStatement: AI-generatedProof: AI-generatedprecheck passaudited 2026-08-11

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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 word-quotient and reduced-word models are uniquely isomorphic compatibly with $X$

Statement

Let $F_{r e d} (X)$ be the reduced-word group of Reduced words form the free group on an alphabet. There is a unique group isomorphism

$Φ : F_{w o r d} (X) ⟶ F_{r e d} (X)$

such that $Φ ([x]) = x$ for every $x \in X$ . It sends each word class to its unique reduced representative, so the quotient-of-words and reduced-word constructions are compatible models of the same free group rather than rival definitions.

Facts & Assumptions

Given: A set $X$ , the word-quotient free group, and the reduced-word free group on $X$ .

[L1]

If $(F, i)$ and $(F^{'}, i^{'})$ are free groups on the same set $X$ , then there is a unique group isomorphism $ϕ : F \to F^{'}$ compatible with the two generator maps (Free groups on the same set are uniquely isomorphic compatibly with their generators).

[L2]

Reduced words form a group whose product is concatenation followed by free reduction, and the map sending $x \in X$ to the one-letter word $x$ has the universal property of the free group on $X$ (Reduced words form the free group on an alphabet).

[L3]

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

[L4]

The word-quotient group together 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 $X$ ).

Proof

technique · direct

1.1

By [L4] and [L2], both displayed models are free groups on the same set $X$ , so [L1] gives a unique compatible isomorphism $Φ : F_{w o r d} (X) \to F_{r e d} (X)$ .

L1L2L4

2.1

Compatibility gives $Φ ([x]) = x$ , and preservation of inverses gives $Φ ([x^{- 1}]) = x^{- 1}$ . Thus $Φ ([a_{1} \dots a_{n}])$ is the reduced product of the one-letter words $a_{1}, \dots, a_{n}$ . It is freely equivalent to $a_{1} \dots a_{n}$ and hence is the unique reduced representative of that class by [L3], including the empty class.

step 1.1L2L3∎

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The word-quotient and reduced-word models are uniquely isomorphic compatibly with XX