TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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In a C prime(1/6) group, every nontrivial torsion element is conjugate to a power of a relator root

Statement

Let $G = ⟨ X ∣ R ⟩$ be a symmetrised $C^{'} (1 / 6)$ presentation. Every nontrivial torsion element of $G$ is conjugate to a power of a root of some defining relator.

Facts & Assumptions

Given: A nontrivial torsion element $g \in G$ .

[F1]

Powers in a group are written multiplicatively as in Powers $g^{n}$ : natural exponents in a monoid and integer exponents in a group, with $g^{0} = e$ .

[F2]

Theorem 5.6 of the cited Williams source is the classical torsion theorem for symmetrised $C^{'} (1 / 6)$ presentations: every nontrivial element of finite order is conjugate to a power of a root of some defining relator.

Proof

technique · direct

1.1

Because $g$ is a nontrivial torsion element, the hypotheses of [F2] apply directly. Therefore $g$ is conjugate to a power of a root of some defining relator.

F2given

2.1

Powers are interpreted as in [F1], so step 1.1 is exactly the claimed conclusion.

F1step 1.1∎

Depends on

Powers $g^{n}$: natural exponents in a monoid and integer exponents in a group, with $g^{0} = e$

Used by

A C prime(1/6) presentation with no proper-power relators defines a torsion-free group Corollary

Dependency tree · two levels

12 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)