Kourovka Notebook Problem 14.3

Definitions and mathematical statement

Let $G$ be a finite group, and let ${\mathbb{Z}}[G]$ be its integral group ring, whose elements are sums $a=\sum_{g\in G}a_g g$ with $a_g\in{\mathbb{Z}}$, with multiplication induced by the group multiplication and distributivity. Write ${\mathbb{Z}}[G]^\times$ for its group of units and $Z(G)$ for the center of $G$. A unit $u\in{\mathbb{Z}}[G]^\times$ is central if $ua=au$ for every $a\in{\mathbb{Z}}[G]$. Define the canonical antiinvolution by \[ \left(\sum_{g\in G}a_g g\right)^*=\sum_{g\in G}a_g g^{-1}. \] Thus $(ab)^*=b^*a^*$. A unit $v$ is symmetric if $v^*=v$.

The question is whether, for every finite group $G$ and every central unit $u\in{\mathbb{Z}}[G]^\times$, there exist $z\in Z(G)$ and a central unit $v\in{\mathbb{Z}}[G]^\times$ such that \[ u=zv\qquad\text{and}\qquad v^*=v, \] where $z$ is identified with its basis element in ${\mathbb{Z}}[G]$. No augmentation restriction on the units is stated in the supplied record.

Short English statement

Does every central unit of the integral group ring of every finite group factor as a central group element times a symmetric central unit, where symmetry means invariance under inversion of the group basis elements?

Sources

• Supplied local UnsolvedMath registry record: permanent ID $700988$, exact source problem code catalogue entry, title Kourovka Notebook Problem 14.3; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Attributed to R. Zh. Aleev, proposed in 1999. The supplied source identifies The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), Problem 14.3, and the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The registry labels the problem open and records an update timestamp of 2026-10-16T14:00:00Z. This is supplied registry information, not an independently verified assessment of its mathematical status or the literature.