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

Abstract

For which groups G is it true that for all fields k, every non-monomial element of the group algebra $kG$ generates a proper 2-sided ideal? The only groups for which we know this to be true are the torsion-free abelian groups. We would, in particular, like to know whether it is true for all free groups. We show that the above property fails for wide classes of groups: for every group G that contains an element $g\ne 1$ whose image in $G/[g,G]$ has finite order (in particular, every group containing a $g\ne 1$ that has finite order, or that satisfies $g\in [g,G]);$ and for every group containing an element g which commutes with a distinct conjugate $h^{-1}gh\ne g$ (in particular, for every nonabelian solvable group). Closure properties of the class of groups satisfying the desired condition are noted. Further questions are raised. In particular, a plausible Freiheitssatz for group algebras of free groups is stated, which would imply the hoped-for result for such group algebras.

Keywords:

• Free group
• group algebra
• 2-sided ideal generated by a non-monomial element

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 20C07
• Secondary: 03C20
• 20E05
• 20F16
• 20F36

Acknowledgments

I am indebted to Dave Witte Morris for correspondence discussing Question 2, to Warren Dicks, Peter Linnell, Don Passman, Tom Scanlon and Eddy Godelle respectively for helping me with references and related material used in Theorem 25, Proposition 9, Section 5, Section 4, and the paragraph at the end of Section 5; and, finally, to the referees of several versions of this note for very helpful comments and pointers to the literature.

Disclosure statement

No potential conflict of interest was reported by the author(s).