A counterexample for the isomorphism-problem of polycyclic groups. (English) Zbl 0838.20005
The authors construct two non-isomorphic polycyclic groups \(G_1\) and \(G_2\) such that the integral group rings \(\mathbb{Z} G_1\) and \(\mathbb{Z} G_2\) are Morita equivalent. Hence for a suitable ring \(R\) of algebraic integers in a global algebraic number field, \(RG_1\) is isomorphic to \(RG_2\). This result gives an answer to an open problem in the representation theory of groups.
Reviewer: Dinh van Huynh (Lima / Ohio)
MSC:
| 20C07 | Group rings of infinite groups and their modules (group-theoretic aspects) |
| 20F16 | Solvable groups, supersolvable groups |
| 16S34 | Group rings |
Keywords:
isomorphism problem; Morita equivalent integral group rings; ring of integers; polycyclic groups; algebraic number fieldReferences:
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| [2] | Roggenkamp, K. W.; Scott, L. L., Isomorphisms of \(p\)-adic group rings, Ann. of Math., 126, 593-647 (1987) · Zbl 0633.20003 |
| [3] | Roggenkamp, K. W.; Zimmerman, A., Outer group automorphisms may become inner in their integral group ring, J. Pure Appl. Algebra, 103, 91-99 (1995), (this issue) · Zbl 0835.16020 |
| [4] | Scott, L. L., Defect groups and the isomorphism problem; Représentations linéaires des groupes finis, (Proc. Colloq. Luminy. Proc. Colloq. Luminy, Fr. 1988. Proc. Colloq. Luminy. Proc. Colloq. Luminy, Fr. 1988, Astérisque, 181-182 (1990)) · Zbl 0727.20002 |
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