TheoremStatement: Literature-sourcedProof: AI-generatedverified 2026-09-23 (gpt-6-sol)

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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]

The published local theorem C prime one sixth torsion elements come from relator roots proves that a nonidentity finite-order element in the symmetrised $C^{'} (1 / 6)$ convention of Sc toolkit symmetrised relators and pieces is conjugate to a power of a root of a cyclic conjugate of a defining relator. This convention uses nonempty cyclically reduced words and permits proper powers and infinite relator sets.

Proof

technique · direct

1.1F2given

Because $g$ is a nontrivial torsion element in the stated symmetrised $C^{'} (1 / 6)$ presentation, [F2] applies. It supplies a cyclic conjugate $r^{'} = v u$ of a defining word $r = u v$ and a word root $c$ of $r^{'}$ such that $g$ is conjugate to a power of the image of $c$ . In the symmetrised relator set, $r^{'}$ is itself a defining relator; equivalently $r^{'}$ and $r$ are conjugate in the free group. Thus this is a power of a root of a defining relator as claimed.

2.1F1step 1.1∎

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

Depends on

Used by

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

Dependency tree · two levels

18 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)