[#P2854] A support-three zero divisor over a torsion-free group

A flat mathematical diagram showing three supported group-ring terms entering a product. — A schematic view of three supported group-ring terms entering a product.

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Problem. Does there exist a torsion-free group $G$ and nonzero elements $\alpha,\beta\in\mathbb F_2[G]$ such that $\alpha\beta=0$ and $\lvert\operatorname{supp}(\alpha)\rvert=3$?

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Definitions and notation

1Context

The support restriction turns the broad zero-divisor conjecture into a sharply constrained combinatorial case. Enumerated cancellation graphs, presentations, and partner-support exclusions are portable artifacts for distributed searches.

2Problem setup

Definition 1 (The group algebra $\mathbb F_2[G]$ consists of finite formal sums $\sum_{g\in G}c_g g$ with $c_g\in\mathbb F_2$, multiplied using the group law and distributivity). The group algebra $\mathbb F_2[G]$ consists of finite formal sums $\sum_{g\in G}c_g g$ with $c_g\in\mathbb F_2$, multiplied using the group law and distributivity.

Definition 2 (The support of $\alpha=\sum c_g g$. The support of $\alpha=\sum c_g g$ is $\{g\in G:c_g\ne0\}$.

Definition 3 (A group). A group is torsion-free if its identity is the only element of finite order.

Definition 4 (A zero-divisor pair here requires both $\alpha\ne0$ and $\beta\ne0$. A zero-divisor pair here requires both $\alpha\ne0$ and $\beta\ne0$.

Remark 1. The support restriction turns the broad zero-divisor conjecture into a sharply constrained combinatorial case. Enumerated cancellation graphs, presentations, and partner-support exclusions are portable artifacts for distributed searches.

3What counts as a solution

For a positive answer, give a finite or recursive presentation of a torsion-free group $G$, explicit finite supports and coefficients for nonzero $\alpha,\beta\in\mathbb F_2[G]$, and verify $\alpha\beta=0$ in the group algebra.
For a negative answer, prove that every support-three element of $\mathbb F_2[G]$ is regular for every torsion-free group $G$.

1Status

What counts as a solution

Current status (Current status and unresolved remainder). UNKNOWN as of 2026-07-31. Abdollahi and Taheri report support size three over $\mathbb F_2$ as the first unsettled case of Kaplansky's zero-divisor conjecture and prove lower bounds on the support of a partner. Recent work proves the conjecture for further group classes, while the general case remains open in the checked sources. For a positive answer, give a finite or recursive presentation of a torsion-free group $G$, explicit finite supports and coefficients for nonzero $\alpha,\beta\in\mathbb F_2[G]$, and verify $\alpha\beta=0$ in the group algebra.^[1]

1Packet records

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Record	Kind	Assessment
UNKNOWN as of 2026-07-31. Abdollahi and Taheri report support size three over $\mathbb F_2$ as the first unsettled case of Kaplansky's zero-divisor conjecture and prove lower bounds on the support of a partner. Recent work proves the conjecture for further group classes, while the general case remains open in the checked sources. For a positive answer, give a finite or recursive presentation of a torsion-free group $G$, explicit finite supports and coefficients for nonzero $\alpha,\beta\in\mathbb F_2[G]$, and verify $\alpha\beta=0$ in the group algebra.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 UNKNOWN as of 2026-07-31. Abdollahi and Taheri report support size three over $\mathbb F_2$ as the first unsettled case of Kaplansky's zero-divisor conjecture and prove lower bounds on the support of a partner. Recent work proves the conjecture for further group classes, while the general case remains open in the checked sources. For a positive answer, give a finite or recursive presentation of a torsion-free group $G$, explicit finite supports and coefficients for nonzero $\alpha,\beta\in\mathbb F_2[G]$, and verify $\alpha\beta=0$ in the group algebra.^[1] Relevance to this problem For support three zero divisor f2 group ring, pins the dated research frontier: UNKNOWN as of 2026-07-31. Abdollahi and Taheri report support size three over $\mathbb F_2$ as the first unsettled case of Kaplansky's zero-divisor conjecture and prove lower bounds on the support of a partner. Recent work proves the conjecture for further group classes, while the general case. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details UNKNOWN as of 2026-07-31. Abdollahi and Taheri report support size three over $\mathbb F_2$ as the first unsettled case of Kaplansky's zero-divisor conjecture and prove lower bounds on the support of a partner. Recent work proves the conjecture for further group classes, while the general case remains open in the checked sources. A complete resolution must satisfy this condition: For a positive answer, give a finite or recursive presentation of a torsion-free group $G$, explicit finite supports and coefficients for nonzero $\alpha,\beta\in\mathbb F_2[G]$, and verify $\alpha\beta=0$ in the group algebra. Open full record →
For a positive answer, give a finite or recursive presentation of a torsion-free group $G$, explicit finite supports and coefficients for nonzero $\alpha,\beta\in\mathbb F_2[G]$, and verify $\alpha\beta=0$ in the group algebra.	Trace	Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 For a positive answer, give a finite or recursive presentation of a torsion-free group $G$, explicit finite supports and coefficients for nonzero $\alpha,\beta\in\mathbb F_2[G]$, and verify $\alpha\beta=0$ in the group algebra.^[1] Relevance to this problem For A support-three zero divisor over a torsion-free group, record support-three-zero-divisor-f2-group-ring-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: For a positive answer, give a finite or recursive presentation of a torsion-free group $G$, explicit finite supports and coefficients for nonzero $\alpha,\beta\in\mathbb F_2[G]$, and verify $\alpha\beta=0$ in the group algebra. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. For a positive answer, give a finite or recursive presentation of a torsion-free group $G$, explicit finite supports and coefficients for nonzero $\alpha,\beta\in\mathbb F_2[G]$, and verify $\alpha\beta=0$ in the group algebra. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. UNKNOWN as of 2026-07-31. Abdollahi and Taheri report support size three over $\mathbb F_2$ as the first unsettled case of Kaplansky's zero-divisor conjecture and prove lower bounds on the support of a partner. Recent work proves the conjecture for further group classes, while the general case remains open in the checked sources.

On 2026-07-27 all four MathOverflow answers and their comments were checked. They discuss known group classes and small-support restrictions without constructing a torsion-free counterexample.
Abdollahi and Taheri, arXiv:1905.09494, call support length three the first unsettled case and analyze its zero-divisor graphs. Earlier work, arXiv:1612.00934, proves that a partner $\beta$ must have support at least 20 in this setting.
The normalization $\alpha=1+g+h$ is often available after multiplying by a group element. Any enumeration must still track identifications forced by the product and prove that the resulting group is torsion-free.
A finite presentation and a formally checked torsion-free argument would be reusable even if a candidate product later collapses. Likewise, exclusions by support size or zero-divisor graph type accumulate cleanly.
Trap: a group with torsion gives immediate zero divisors and is excluded. A relation table suggesting torsion-freeness is not a torsion-free certificate.

Recorded example 1. If $G$ contains an element $g$ of order two, then $(1+g)^2=0$ over $\mathbb F_2$; the torsion-free hypothesis removes this elementary source of zero divisors.

How the 2 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemA support-three zero divisor over a torsion-free group

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

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Plain text

“A support-three zero divisor over a torsion-free group.” TheoremDB. P2854. Problem statement; statement text SHA-256 6c07ca5dfc87b4b57923c1c0c3b11968fcd3c1025422d6fc8f41c30e32eeb17f. https://theoremdb.org/statement/?ref=P2854

BibTeX

@misc{theoremdb-problem-6c07ca5dfc87b4b57923c1c0c3b11968fcd3c1025422d6fc8f41c30e32eeb17f,
  title = {{A support-three zero divisor over a torsion-free group}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 6c07ca5dfc87b4b57923c1c0c3b11968fcd3c1025422d6fc8f41c30e32eeb17f},
  url = {https://theoremdb.org/statement/?ref=P2854}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 1 typed links, sourced from mathoverflow.net^[1], current as of July 31, 2026.

1References

Packet source. Zero-divisor conjecture for finite fields, MathOverflow question 62548. Original CC0 minimal-support target written after reading all four answers and comments and checking the support-three literature. mathoverflow.net checked 2026-08-01. Original CC0 minimal-support target written after reading all four answers and comments and checking the support-three literature. ↗forum · discovery source · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. Abdollahi and Taheri report support size three over $\mathbb F_2$ as the first unsettled case of Kaplansky's zero-divisor conjecture and prove lower bounds on the support of a partner. Recent work proves the conjecture for further group classes, while the general case remains open in the checked sources.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For A support-three zero divisor over a torsion-free group: UNKNOWN as of 2026-07-27. Abdollahi and Taheri report support size three over $\mathbb F_2$ as the first unsettled case of Kaplansky's zero-divisor conjecture and prove lower bounds on the support of a partner. Recent work proves the conjecture for further group classes, while the general case remains open in the checked sources.Source named by the research packet.
Alireza Abdollahi and Zahra Taheri, “Zero divisors of support size $3$ in group algebras and trinomials divided by irreducible polynomials over $GF(2)$”. DOI 10.4171/RSMUP/78. arXiv:1905.09494 (2019). Status evidence identified in the source record and checked at the linked publication. ↗preprint · primary source · arXiv:1905.09494, checked 2026-07-31 · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. Abdollahi and Taheri report support size three over $\mathbb F_2$ as the first unsettled case of Kaplansky's zero-divisor conjecture and prove lower bounds on the support of a partner. Recent work proves the conjecture for further group classes, while the general case remains open in the checked sources.Also cited at Full preprint relevant to A support-three zero divisor over a torsion-free group.Source used to assess the problem's recorded status.For A support-three zero divisor over a torsion-free group: UNKNOWN as of 2026-07-27. Abdollahi and Taheri report support size three over $\mathbb F_2$ as the first unsettled case of Kaplansky's zero-divisor conjecture and prove lower bounds on the support of a partner. Recent work proves the conjecture for further group classes, while the general case remains open in the checked sources.
Alireza Abdollahi and Zahra Taheri, “Kaplansky's zero divisor and unit conjectures on elements with supports of size $3$”. arXiv:1612.00934 (2016). Status evidence identified in the source record and checked at the linked publication. ↗preprint · primary source · arXiv:1612.00934, checked 2026-07-31 · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. Abdollahi and Taheri report support size three over $\mathbb F_2$ as the first unsettled case of Kaplansky's zero-divisor conjecture and prove lower bounds on the support of a partner. Recent work proves the conjecture for further group classes, while the general case remains open in the checked sources.Also cited at Full preprint relevant to A support-three zero divisor over a torsion-free group.Source used to assess the problem's recorded status.For A support-three zero divisor over a torsion-free group: UNKNOWN as of 2026-07-27. Abdollahi and Taheri report support size three over $\mathbb F_2$ as the first unsettled case of Kaplansky's zero-divisor conjecture and prove lower bounds on the support of a partner. Recent work proves the conjecture for further group classes, while the general case remains open in the checked sources.
Alireza Abdollahi and Fatemeh Jafari, “Zero divisor and unit elements with support of size 4 in group algebras of torsion free groups”. arXiv:1709.08204 (2017). Status evidence identified in the source record and checked at the linked publication. ↗preprint · primary source · arXiv:1709.08204, checked 2026-07-31 · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. Abdollahi and Taheri report support size three over $\mathbb F_2$ as the first unsettled case of Kaplansky's zero-divisor conjecture and prove lower bounds on the support of a partner. Recent work proves the conjecture for further group classes, while the general case remains open in the checked sources.Also cited at Full preprint relevant to A support-three zero divisor over a torsion-free group.Source used to assess the problem's recorded status.For A support-three zero divisor over a torsion-free group: UNKNOWN as of 2026-07-27. Abdollahi and Taheri report support size three over $\mathbb F_2$ as the first unsettled case of Kaplansky's zero-divisor conjecture and prove lower bounds on the support of a partner. Recent work proves the conjecture for further group classes, while the general case remains open in the checked sources.
Sam P. Fisher and Pablo Sánchez-Peralta, “Division Rings for Group Algebras of Virtually Compact Special Groups and 3-Manifold Groups,” Journal of Combinatorial Algebra 10(1-2) (2026), 153-193. DOI 10.4171/JCA/89. Status evidence identified in the source record and checked at the linked publication. ↗website · primary source · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. Abdollahi and Taheri report support size three over $\mathbb F_2$ as the first unsettled case of Kaplansky's zero-divisor conjecture and prove lower bounds on the support of a partner. Recent work proves the conjecture for further group classes, while the general case remains open in the checked sources.Also cited at main division-ring theorem for virtually compact special groups and 3-manifold groups.Source used to assess the problem's recorded status.For A support-three zero divisor over a torsion-free group, this source proves the zero-divisor conjecture for major group classes without resolving the general support-three case over a torsion-free group.

Original CC0 textbook restatement.

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[#P2854] A support-three zero divisor over a torsion-free group

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3What counts as a solution

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