Hypercentral units in integral group rings (Q2718995)

Let \(G\) be any group, let \(\mathbb{Z} G\) the integral group ring and let \(V(\mathbb{Z} G)\) be the group of units of \(\mathbb{Z} G\) of augmentation 1. Let \(Z_n\) be the \(n\)-th term of the ascending central series of \(V(\mathbb{Z} G)\). Arora, Hales and Passi treated the case of a finite group \(G\) completely. The case of a torsion group \(G\) was started by the first author and in the paper under review this research is continued. The authors prove that for a torsion group \(G\) one has \(Z_2\neq Z_1\) if and only if \(G\) is a so-called \(Q^*\)-group. If \(G\) is a \(Q^*\)-group, then either \(G\) is a Hamilton 2-group, in which case \(Z_2=G\), or \(Z_2=\langle a\rangle\cdot Z_1\) where \(a\) is an element of \(G\) of order 4. A \(Q^*\)-group is a group with very restrictive properties, similar to those of a quaternion group.NEWLINENEWLINENEWLINEThe proof goes by tricky arguments constructing bicyclic, Bass cyclic and Hoechsmann units having certain properties which then restricts the structure of \(G\).