Which group algebras cannot be made zero by imposing a single non-monomial relation?
| Autor: | Bergman, George M. |
|---|---|
| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | Communications in Algebra, 49 (2021) 3760-3776 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1080/00927872.2021.1905825 |
| Popis: | 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. Comment: 14 pp. Final version sent to publisher |
| Databáze: | arXiv |
| Externí odkaz: |
|