The computation of ranks of unit groups of integral group rings of finite groups

Abstract

The study of central units (central invertible elements) of integral group rings is encountered to difficult calculations almost everywhere, both in the case of finding of individual central unit and in the case of describing of group of central elements.
By virtue of torsion part of central unit group is trivial (up to sign those are elements of center group) it is more interesting to find data about torsion free part that is direct product infinite cyclic groups. The number of such infinite factors is the rank of central unit group. Therefore the ranks of central unit groups of integral group rings of finite groups are the very important characteristic those groups.
So that the computation ranks of central unit groups has big interest for study of central unit groups. In the paper we point out the formulas for computation of ranks in general case and some important particular cases. On the base of those formulas we compute the ranks in quite large ranges. We used computer algebra system GAP. The results are shown on tables and graph.