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Kaplansky Direct-Finiteness Conjecture

The Kaplansky direct-finiteness conjecture asserts that the group algebra K[G] of any group G over a field K is directly finite, meaning that ab=1 implies ba=1 for a,b in K[G]. Unlike the Kaplansky zero-divisor conjecture, it does not assume that G is a torsion-free group (Bunina et al. 2026).

The conjecture holds for sofic groups over every field (Elek and Szabó 2004). OpenAI (2026, p. 1) constructed a counterexample in one specified odd field characteristic. For m=(1200; 600), let p be the smallest prime factor of (m!)^2+1. Then p is odd, and there are a finite field K with p^4 elements, a finitely generated group G with torsion, and a,b in K[G] with ab=1 but ba!=1. MathVet (2026) published a successful Lean-kernel check of this construction and its defining identities, using only Lean's standard axioms. This example also refutes the Gottschalk surjunctivity conjecture by giving a cellular automaton that is injective but not surjective on K^G (OpenAI 2026).

Bunina et al. (2026) report that for every odd prime number p there is a finitely presented group G_p with a,b,c in GF(p)[G_p] satisfying ab=1, ac=0, and c!=0. These relations imply ba!=1, since ba=1 would give c=bac=0. The same counterexample persists over any field of field characteristic p. The claimed construction would also provide a group that is not a sofic group.

This broader argument depends on substantial geometric and probabilistic inputs from earlier OpenAI manuscripts. The formal check of OpenAI's specified-prime example does not establish this broader claim.

Mghirbi and Douglas (2026) showed that for each prime number p there is a finitely presented, torsion-free hyperbolic group G with a finite two-dimensional classifying space whose group algebra K[G] contains elements a,b with ab=1!=ba over every field K of field characteristic p. Their characteristic-2 result establishes hyperbolicity for groups from an earlier OpenAI construction. For odd p they construct new groups by adapting its cancellation argument. The authors report that AI models worked out the exact color-balance solution and transferred the random-graph estimates to odd field characteristic. The authors report a Lean formalization of the existence theorem using only standard axioms. The formalization uses the random-graph estimates in existence form and does not establish all of their probabilistic results.

See also

Field Characteristic, Gottschalk Surjunctivity Conjecture, Group Algebra, Kaplansky Zero-Divisor Conjecture, Sofic Group

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References

Bunina, E. I.; Lee, W.; Miasnikov, A.; and Muliarchyk, K. "Failure of Direct Finiteness for Group Algebras in Every Odd Characteristic." 15 Oct 2026. https://arxiv.org/abs/2610.22536.Elek, G. and Szabó, E. "Sofic Groups and Direct Finiteness." J. Algebra 280, 426-434, 2004. https://doi.org/10.1016/j.jalgebra.2004.06.023.MathVet. "OddKaplansky.comparator.txt." Oct. 8, 2026. https://github.com/mathvet/mathvet/blob/8cc0dcf0e63248df1dac8d7ec6a5f2bb5cb9632b/reviews/openai-math/evidence/cloud/OddKaplansky.comparator.txt.Mghirbi, N. and Douglas, S. "Kaplansky's Conjectures Fail for Torsion-Free Hyperbolic Groups in Positive Characteristic." 3 Nov 2026. https://arxiv.org/abs/2611.04599v1.OpenAI. "A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic." Sep. 26, 2026. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026/paper.pdf. Lean formalization: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Algebra/OddKaplansky/Main.lean.

Cite this as:

Weisstein, Eric W. "Kaplansky Direct-Finiteness Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/KaplanskyDirect-FinitenessConjecture.html

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