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

Document Type: Research Paper

Author

University of Stuttgart

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 Al Ain 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 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 $p = 2$ 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

Main Subjects


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