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

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

Statement refuted

No proper-power relators alone prevent torsion, even without any small- cancellation hypothesis.

Facts & Assumptions

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

[L1]

The torsion-free consequence on this page needs the small-cancellation hypothesis as well as the no-proper-power hypothesis (A C prime(1/6) presentation with no proper-power relators defines a torsion-free group).

Counterexample

technique · direct

1.1

As in FALSE: a presentation with no proper-power relators is automatically torsion-free, neither relator $a b^{2}$ nor $a^{2} b$ is a proper power.

given

2.1

The same substitution calculation gives $a = b^{- 2}$ and then $b^{- 4} b = 1$ , so $b^{3} = 1$ . Moreover the assignment $a \mapsto t$ , $b \mapsto t$ to the cyclic group $C_{3} = ⟨ t ∣ t^{3} = 1 ⟩$ satisfies both relators, so it induces a surjective homomorphism $G \to C_{3}$ . Thus $G$ has nontrivial torsion.

step 1.1algebra

3.1

Therefore no-proper-power relators do not by themselves prevent torsion. By [L1], the missing small-cancellation hypothesis is essential.

L1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)