Sylow like theorems for V(ZG)

Abstract

The main part of this article is a survey on torsion subgroups of the unit group of an integral group ring. It contains the major parts of my talk given at the conference "Groups, Group Rings and Related Topics" at UAEU in AlAin October 2013. In the second part special emphasis is layed on p -subgroups and on the open question whether there is a Sylow like theorem in the normalized unit group of an integral group ring. For speci c classes of nite groups we prove that p -subgroups of the normalized unit group of its integral group rings V(ZG) are isomorphic to subgroups of G : In particular for p=2 this is shown when G has abelian Sylow 2 -subgroups. This extends results known for soluble groups to classes of groups which are not contained in the class of soluble groups.

Keywords

• Group rings
• Integral group ring
• Subgroup isomorphism problem