False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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FALSE: a presentation with no proper-power relators is automatically torsion-free

Statement

A presentation with no proper-power relators is automatically torsion-free.

Facts & Assumptions

Given: The presentation $G = ⟨ a, b ∣ a b^{2}, a^{2} b ⟩$ .

[L1]

The torsion-free conclusion on this page needs both the $C^{'} (1 / 6)$ hypothesis and the no-proper-power hypothesis (A C prime(1/6) presentation with no proper-power relators defines a torsion-free group).

[F1]

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

Refutation

technique · direct

1.1

Neither relator $a b^{2}$ nor $a^{2} b$ is a proper power: each is cyclically reduced of length $3$ and is not a repetition of a shorter cyclic word.

given

2.1

From $a b^{2} = 1$ one gets $a = b^{- 2}$ , and substituting this into $a^{2} b = 1$ gives $b^{- 4} b = 1$ , hence $b^{3} = 1$ in the sense of [F1]. Moreover, if $C_{3} = ⟨ t ∣ t^{3} = 1 ⟩$ , then the assignment $a \mapsto t$ , $b \mapsto t$ satisfies both relators, so it induces a surjective homomorphism $G \to C_{3}$ . Therefore the image of $b$ is nontrivial and $G$ contains a nontrivial torsion element.

F1step 1.1algebra

3.1

Therefore the absence of proper-power relators alone does not force torsion-freeness. By [L1], the missing small-cancellation hypothesis is load-bearing.

L1step 2.1∎

Depends on

Used by

No proper-power relators alone do not prevent torsion outside small cancellation Counterexample

Dependency tree · two levels

14 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)