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

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

Statement

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

$\Phi:F_{\mathrm{word}}(X)\longrightarrow F_{\mathrm{red}}(X)$

such that $\Phi([x])=x$ 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 $\phi:F\to F'$ 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 $\Phi:F_{\mathrm{word}}(X)\to F_{\mathrm{red}}(X)$ .

L1L2L4

2.1

Compatibility gives $\Phi([x])=x$ , and preservation of inverses gives $\Phi([x^{-1}])=x^{-1}$ . Thus $\Phi([a_1\cdots a_n])$ is the reduced product of the one-letter words $a_1,\ldots,a_n$ . It is freely equivalent to $a_1\cdots a_n$ and hence is the unique reduced representative of that class by [L3], including the empty class.

step 1.1L2L3∎

Depends on

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