Zero Divisors in a Seven-Generator Group Ring

Definitions and mathematical statement

Let \[ \Delta=\langle a,b,c,d,e,f,g\mid acb=1,\ add=1,\ afg=1,\ bbf=1,\ cdg=1,\ cef=1,\ eeg=1\rangle. \] The supplied record describes $\Delta$ as torsion-free and as lacking the unique-product property. A group $G$ is torsion-free if $g^n=1$, for $g\in G$ and an integer $n\geq 1$, implies $g=1$. It has the unique-product property if, for every pair of finite nonempty subsets $A,B\subseteq G$, some element of $AB$ has exactly one representation $ab$ with $(a,b)\in A\times B$. These descriptions of $\Delta$ are supplied source assertions, not independently established here.

For a field $K$, the group algebra $K[\Delta]$ consists of finite formal sums $x=\sum_{g\in\Delta}x_g g$, with coefficients $x_g\in K$. Addition is coefficientwise, and multiplication is the bilinear extension of multiplication in $\Delta$: \[ \left(\sum_g x_g g\right)\left(\sum_h y_h h\right) =\sum_{t\in\Delta}\left(\sum_{gh=t}x_g y_h\right)t. \] The question is whether $\Delta$ satisfies Kaplansky's zero-divisor conjecture over every field; explicitly, whether \[ \text{for every field }K\text{ and all }x,y\in K[\Delta], \qquad xy=0\ \Longrightarrow\ x=0\text{ or }y=0. \] There is no restriction on the characteristic of $K$.

Supplied contextual qualifications. The original passage says that the unit conjecture remains open in characteristic zero, that Murray subsequently obtained counterexamples using the same group $P$ in every positive characteristic (its reference [6]), and that the zero-divisor conjecture is known for many groups, including residually torsion-free elementary amenable groups (its reference [5]). It contrasts this with failure of the unit conjecture for the Hantzsche--Wendt group. It also states that failure of the zero-divisor conjecture would yield a counterexample in positive characteristic, and that, for torsion-free $G$ and a field $K$ of positive characteristic, the conjecture for $K[G]$ is equivalent to its unit group having "no non-trivial torsion." These are retained as supplied contextual claims, not independently verified facts. The passage does not define $P$, identify references [5] and [6] bibliographically, or specify the meaning of "non-trivial torsion." In particular, the latter phrase cannot simply mean absence of all nonidentity finite-order units, since scalar units can have finite order. This unresolved wording is not used to replace the explicit zero-divisor question above.

Short English statement

For the specified seven-generator group $\Delta$, ask whether the product of two nonzero elements of $K[\Delta]$ is nonzero for every field $K$, in every characteristic. The supplied record labels the problem open; this statement-only draft does not independently verify that status or the contextual literature claims.

Sources

• Local UnsolvedMath registry, supplied complete record with permanent ID $30005037$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2022 and names no proposer.
• Supplied source citation: Geometric Structures in Group Theory (2023), Oberwolfach Reports, DOI 10.4171/owr/2022/11, pp. 517--576. https://doi.org/10.4171/owr/2022/11. The background attributes the problem to workshop open-problem discussions documented in this report.
• Supplied literature source: On zero divisors and units in group rings. https://arxiv.org/abs/2501.07646. No authors or further bibliographic details for this item are supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels this question open and says that the general zero-divisor conjecture remains unresolved. This is supplied dated curation, not an independently verified assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied record labels the statement corrected_verified and reports that the official Oberwolfach report was checked directly to restore definitions and remove extraction artifacts. This verification claim belongs to the supplied record; no source visit or independent verification is claimed here. The original passage's references [5] and [6] are supplied without their bibliography entries.