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

- ID: `P2854`
- Reference: `support-three-zero-divisor-f2-group-ring`
- Page: https://theoremdb.org/statements/P2854
- Record maturity: Reviewed problem with recorded work

## 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 \(\lvert\operatorname{supp}(\alpha)\rvert=3\)?

### Problem setup

- **Definition.** 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.
- **Remark.** The support of \(\alpha=\sum c_g g\) is \(\{g\in G:c_g\ne0\}\).
- **Definition.** A group is torsion-free if its identity is the only element of finite order.
- **Remark.** A zero-divisor pair here requires both \(\alpha\ne0\) and \(\beta\ne0\).

### What 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 \(G\).

## 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. 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](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

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.

- Original intake status: 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.
- 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: 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.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `support-three-zero-divisor-f2-group-ring`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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. https://mathoverflow.net/questions/62548/zero-divisor-conjecture-for-finite-fields
   - Also cited at See dataset.references[0] for the exact external source and locator.
   - Also cited at Editorial research route recorded 2026-07-31
   - forum; reference source; checked 2026-07-31
   - Source use: citation_only
   - 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.
2. <a id="reference-2"></a>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). Full preprint relevant to A support-three zero divisor over a torsion-free group. https://arxiv.org/abs/1905.09494
   - preprint; reference source; arXiv:1905.09494, checked 2026-07-31; checked 2026-07-31
   - Source use: citation_only
   - 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.
3. <a id="reference-3"></a>Alireza Abdollahi and Zahra Taheri, “Kaplansky's zero divisor and unit conjectures on elements with supports of size $3$”. arXiv:1612.00934 (2016). Full preprint relevant to A support-three zero divisor over a torsion-free group. https://arxiv.org/abs/1612.00934
   - preprint; reference source; arXiv:1612.00934, checked 2026-07-31; checked 2026-07-31
   - Source use: citation_only
   - 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.
4. <a id="reference-4"></a>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). Full preprint relevant to A support-three zero divisor over a torsion-free group. https://arxiv.org/abs/1709.08204
   - preprint; reference source; arXiv:1709.08204, checked 2026-07-31; checked 2026-07-31
   - Source use: citation_only
   - 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.
5. <a id="reference-5"></a>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. main division-ring theorem for virtually compact special groups and 3-manifold groups https://ems.press/content/serial-article-files/52395
   - website; reference source; checked 2026-07-31
   - Source use: citation_only
   - 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.