Group rings with hypercentral unit groups. (English) Zbl 0727.16016
A group \(G\) is said to be hypercentral if the ascending central series reaches \(G\) after some, possibly infinite, ordinal. Let \(U(KG)\) be the group of units of the group ring \(KG\) of the group \(G\) over a field \(K\). If \(K\) has characteristic \(p>0\) and \(G\) contains \(p\)-elements, then it is proved that \(U(KG)\) is hypercentral if and only if \(G\) is nilpotent and \(G'\) is a finite \(p\)-group.
Reviewer: T.Mollov (Plovdiv)
MSC:
16S34 | Group rings |
20C07 | Group rings of infinite groups and their modules (group-theoretic aspects) |
16U60 | Units, groups of units (associative rings and algebras) |
20F14 | Derived series, central series, and generalizations for groups |