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On the Structure of the Group of Units os the Group Ring of a Torsion Group
by
A.A. Bovdi
Institute of Mathematics and Informatics, Kossuth University, Hungary
Coauthors: C.Polcino Milies
Let RG be the group ring of a torsion group G over an integral domain R. We denote by U(RG) the group of units of the group ring RG and by V(RG) the subgroup of its normalized units. Group algebras have been characterized under different group-theoretical assumption on its group of units, such as being nilpotent, solvable or n-Engel. One of the hardest and most important problem for modular group algebras consists in describing the structure of its group of units.
First Coleman and Passman proved that if FpG is a group algebra o f a finite nonabelian p-group G over the field Fp of p elements, then the group of normalized units V(FpG) is not p-regular group and has exponent at least p2. Clearly, for this algebras V(FpG) = 1+I Fp(G), where IFp(G) is the augmentation ideal of FpG. Then, the adjoint group (IFp(G), o ) is isomorphic to V( FpG) and thus the k-th term \zetak(V) of the upper central series of V( FpG) coincides with 1+Ck, where Ck is both a subring and a Lie ideal of FpG and, at the same time, the k-th term of the upper central series of the associated Lie algebra of the augmentation ideal IFp(G). It follows that the factors \zetai+1(V)/\zetai(V) of the upper central series with the exception of the center of V(FpG) are elementary abelian p-groups.
We extend this results for all normal subgroup of the nilpotent group V( FpG). Moreover, this is true for all nilpotent normal subgroup in V(RG) with weaker assumptions about the group G, namely, for any torsion group G with some restriction on RG.
We also study conjugacy classes of the nilpotent group of normalized units V( FG). Coleman obtained that if G is a finite p-group, Cu is a conjugacy class in V(FpG) and Cu contains an element from G then Cu \cap G is a conjugacy class in G. We prove that there exists a conjugacy class containing no element from G, showing that the conjecture of Zassenhaus does not hold for modular group rings. We impove also a result of Rao and Sandling which states that p2 divides the order |Ca| for every noncentral element a.
Date received: November 9, 1998
Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabw-11.