PropositionStatement: AI-adaptedProof: 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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In $\langle X\mid R\rangle$ , the words $u$ and $v$ represent the same element if and only if $u^{-1}v\in\langle\!\langle R\rangle\!\rangle$

Statement

Let $u,v\in F(X)$ and put $N=\langle\!\langle R\rangle\!\rangle_{F(X)}$ . The words $u$ and $v$ represent the same element of $\langle X\mid R\rangle$ if and only if

$u^{-1}v\in\langle\!\langle R\rangle\!\rangle_{F(X)}.$

By The normal closure of $R$ is the set of finite products of conjugates of elements of $R$ and their inverses, the membership condition is equivalent to expressing $u^{-1}v$ as a finite product of conjugates of relators and their inverses.

Facts & Assumptions

Given: A presentation $\langle X\mid R\rangle$ and words $u,v\in F(X)$ .

[F1]

$\langle X\mid R\rangle=F(X)/\langle\!\langle R\rangle\!\rangle_{F(X)}$ (Group presentation by generators and relations).

[F2]

If $N\mathrel{\trianglelefteq}G$ , then the elements of $G/N$ are the left cosets $gN$ (The quotient group $G/N$ and coset product $(gN)(hN)=ghN$ ).

[L1]

For a subgroup $H$ of a group, $aH=bH$ if and only if $a^{-1}b\in H$ ( $x\in aH$ iff $a^{-1}x\in H$ , and $aH=bH$ iff $a^{-1}b\in H$ ).

Proof

technique · direct

1.1

Set $N=\langle\!\langle R\rangle\!\rangle_{F(X)}$ ; by [F1] and [F2], the elements represented by $u$ and $v$ are the quotient cosets $uN$ and $vN$ .

F1F2given

2.1

By [L1], $uN=vN$ if and only if $u^{-1}v\in N$ .

L1step 1.1

3.1

Substituting the definition of $N$ into step 2.1 proves both directions of the stated equivalence.

step 2.1∎

Depends on

Used by

Two finite presentations define isomorphic groups if and only if a finite sequence of Tietze transformations and inverses connects them Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 25 results over 13 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

Nicholas Touikan, An Introduction to Combinatorial and Geometric Group Theory, §1.4 (standard reference, not scraped)

In ⟨X∣R⟩\langle X\mid R\rangle⟨X∣R⟩, the words uuu and vvv represent the same element if and only if u−1v∈⟨ ⁣⟨R⟩ ⁣⟩u^{-1}v\in\langle\!\langle R\rangle\!\rangleu−1v∈⟨⟨R⟩⟩