# Which group algebras cannot be made zero by imposing a single non-monomial relation?

## TL;DR

This paper investigates which groups have the property that every non-monomial element in their group algebra generates a proper ideal, showing that this property fails for many classes of groups beyond torsion-free abelian groups.

## Contribution

It identifies classes of groups where the property fails and explores closure properties of groups satisfying this condition, raising questions about free groups and related conjectures.

## Key findings

- Property fails for groups with elements of finite order or certain conjugation properties.
- Torsion-free abelian groups are the only known groups with the property.
- Open questions about free groups and a possible Freiheitssatz for their group algebras.

## Abstract

For which groups $G$ is it true that for all fields $k$, every non-monomial element of the group algebra $k\,G$ generates a proper $2$-sided ideal? The only groups for which we know this are the torsion-free abelian groups. We would like to know whether it also holds for all free groups. It is shown that the above property fails for wide classes of groups: for every group $G$ that contains an element $g\neq 1$ whose image in $G/[g,G]$ has finite order (in particular, every group containing a $g\neq 1$ that itself has finite order, or that satisfies $g\in [g,G])$; and for every group containing an element $g$ which commutes with a conjugate $hgh^{-1}\neq g$ (in particular, for every nonabelian solvable group). Results are obtained on closure properties of the class of groups satisfying the stated condition. Many further questions are raised; in particular, a plausible Freiheitssatz for group algebras of free groups is noted.