Sylow like theorems for V(\mathbb{Z}G)

‎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 Al Ain October ۲۰۱۳‎. ‎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 specific classes of finite groups we prove that p‎ - ‎subgroups‎ ‎of the normalized unit group of its integral group rings V(\mathbb{Z}G) ‎are isomorphic to subgroups of G‎ .‎ In particular for p = ۲ this is shown ‎when G has abelian Sylow ۲‎ - ‎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