LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

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.

A relator set and its symmetrisation have the same normal closure

Statement

Let $R$ be a cyclically reduced relator set and let $R^{sym}$ be its symmetrisation. Then $R$ and $R^{sym}$ have the same normal closure in the free group on the generators.

Facts & Assumptions

Given: A cyclically reduced relator set $R$ in a free group $F (X)$ , and its symmetrisation $R^{sym}$ .

[F1]

The normal closure of a subset $S \subseteq F (X)$ is the smallest normal subgroup of $F (X)$ containing $S$ (The normal closure of a subset of a group).

[L1]

Every element of $R^{sym}$ is either a cyclic conjugate of a member of $R$ or of its inverse (The symmetrisation of a relator set closes under inverses and cyclic conjugates).

Proof

technique · direct

1.1F1L1given

Let $N = ⟨ ⁣ ⟨ R ⟩ ⁣ ⟩$ . Because $N$ is normal by [F1], it contains $r^{- 1}$ whenever it contains $r$ , and it contains $u r u^{- 1}$ for every $u \in F (X)$ . Hence [L1] implies that every element of $R^{sym}$ already lies in $N$ . Therefore $⟨ ⁣ ⟨ R^{sym} ⟩ ⁣ ⟩ \subseteq N$ .

1.2F1L1given

Every relator of $R$ belongs to $R^{sym}$ by definition, so the normal closure of $R^{sym}$ contains $R$ . By the minimality clause of [F1], $N \subseteq ⟨ ⁣ ⟨ R^{sym} ⟩ ⁣ ⟩$ .

2.1step 1.1step 1.2∎

The two containments from steps 1.1 and 1.2 are equalities, so the normal closures agree.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

GAP SmallCancellation manual, Chapter 1: Small Cancellation Theory — the classical conditions (standard reference, not scraped)
Jay Williams, Universal Countable Borel Quasi-Orders (standard reference, not scraped)
Nicholas Touikan, An Introduction to Combinatorial and Geometric Group Theory, Section 3.5 (standard reference, not scraped)
Clara Löh, Geometric Group Theory: An Introduction, Section 7.4.1 (standard reference, not scraped)