On hypercentral units in integral group rings. (English) Zbl 1124.16023
Let \(G\) be a group. An integral domain of characteristic \(0\) is called \(G\)-adapted if whenever \(G\) has an element of prime order \(p\), then \(p\) is not invertible in \(R\). Let \(R\) be such a \(G\)-adapted ring and let \(U\) be the group of units of the group ring \(RG\). Let \(Z_n(U)\) be the \(n\)-th term of the upper central series of \(U\), and let \(Z_\infty(U)\) be the union of all the \(Z_n(U)\).
The main result of the paper under review is the statement that all elements in \(Z_\infty(U)\) normalize \(G\). The paper contains many intermediate results which are interesting in their own right. Some of them come from non published parts of the first author’s Habilitationsschrift. Moreover, in case some additional hypotheses are satisfied, then a more detailed structure theory is developed.
The main result of the paper under review is the statement that all elements in \(Z_\infty(U)\) normalize \(G\). The paper contains many intermediate results which are interesting in their own right. Some of them come from non published parts of the first author’s Habilitationsschrift. Moreover, in case some additional hypotheses are satisfied, then a more detailed structure theory is developed.
Reviewer: Alexander Zimmermann (Amiens)
MSC:
| 16U60 | Units, groups of units (associative rings and algebras) |
| 20C07 | Group rings of infinite groups and their modules (group-theoretic aspects) |
| 20F14 | Derived series, central series, and generalizations for groups |
| 16S34 | Group rings |
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