Browsing by Author "Bächle, Andreas"

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On torsion subgroups and their normalizers in integral group rings
(2012) Bächle, Andreas; Kimmerle, Wolfgang (apl. Prof. Dr.)
In view of the Zassenhaus conjectures we show that p-subgroups of the normalized units of integral group rings of p-constrained groups are rationally conjugate to subgroups of a group basis, extending a known result. Moreover, we prove that the corresponding statement holds for 2-subgroups, given that the group basis has abelian Sylow 2-subgroups of order at most 8. We provide an affirmative answer for the prime graph question for the groups SL(2, q), q an odd prime power. The 'classical' normalizer problem asks, if a group basis is normalized in the unit group of the integral group ring by products of group elements and central units. After an overview of known results we consider the corresponding question for subgroups of a group basis. We obtain a positive answer for certain isomorphism types of subgroups, comprising e.g. all cyclic groups, and for certain types of normal subgroups. Considering, for a fixed group basis, the question if there is an affirmative answer to the normalizer problem for all its subgroups we provide a positive answer for all locally nilpotent torsion groups and certain metacyclic groups. The last chapter deals with centralizers of subgroups of a group basis in the unit group of an integral group ring. Besides results dealing directly with the centralizers we use the methods of this chapter to prove that the prime graph of the normalizer of an isolated subgroup (of a finite group) in the group and in the normalized unit group of an integral group ring coincide.