PropositionStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-11

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

In $⟨ X ∣ R ⟩$ , the words $u$ and $v$ represent the same element if and only if $u^{- 1} v \in ⟨ ⁣ ⟨ R ⟩ ⁣ ⟩$

Statement

Let $u, v \in F (X)$ and put $N = ⟨ ⁣ ⟨ R ⟩ ⁣ ⟩_{F (X)}$ . The words $u$ and $v$ represent the same element of $⟨ X ∣ R ⟩$ if and only if

$u^{- 1} v \in ⟨ ⁣ ⟨ R ⟩ ⁣ ⟩_{F (X)} .$

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 $⟨ X ∣ R ⟩$ and words $u, v \in F (X)$ .

[F1]

$⟨ X ∣ R ⟩ = F (X) / ⟨ ⁣ ⟨ R ⟩ ⁣ ⟩_{F (X)}$ (Group presentation by generators and relations).

[F2]

If $N ⊴ G$ , then the elements of $G / N$ are the left cosets $g N$ (The quotient group $G / N$ and coset product $(g N) (h N) = g h N$ ).

[L1]

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

Proof

technique · direct

1.1

Set $N = ⟨ ⁣ ⟨ R ⟩ ⁣ ⟩_{F (X)}$ ; by [F1] and [F2], the elements represented by $u$ and $v$ are the quotient cosets $u N$ and $v N$ .

F1F2given

2.1

By [L1], $u N = v N$ 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

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

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

In ⟨X∣R⟩⟨X∣R⟩, the words uu and vv represent the same element if and only if u−1v∈⟨ ⁣⟨R⟩ ⁣⟩u−1v∈⟨⟨R⟩⟩