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1981, Proceedings of the American Mathematical Society
Indian Journal of Pure and Applied Mathematics
Structure of group rings and the group of units of integral group rings: an invitation2021 •
2002 •
We present a survey of some recent results on problems posed by Sudarshan Sehgal. In this paper some new results are presented on the following problems posed by Sudarshan Sehgal in [43] . The integral group ring of a group G is denoted by ZG. Its unit group we denote by U(ZG), its group of normalized units by UI (ZG) and its center by Z(U(ZG» . Problem 17: Give a presentation by generators and relations for the unit group U(ZG) of the integral group ring ZG for some finite groups G . Problem 23: Give generators up t.o finite index for U(ZG) if G is a finite group. Problem 18: Find a good estimate for the index (U1 (7l.. G) : B) , where B is the group generated by the Bass cyclic units and the bicyclic units. Problem 19: Is the group generated by the bicylic units torsion-free? Problem 29: Suppose that G is a finite nilpotent group . Does G have a normal complement N in UI (Z G) , i.e., UI (ZG) = N >4 G? Problem 43: Let G be a finite group. Is Nu(zG)(G) = G Z(U(ZG))? In the late ...
This thesis deals with the problem of describing the unit group of specific group rings over the integers. Some results are given in an expository manner as the proofs of the results are normally very dependent on the particular group. The ones presented by this method later on are S3,D4,D6 and A4. Next, we present the method for groups of order p3, where p is an odd prime. These come from a paper by Ritter and Sehgal. I consider both non-abelian groups of order p3 and descriptions of the unit groups of both of their respective group rings are presented.I present the method as applied to the groups of order 27. The last theoretical results are on determining the unit group of the group ring over a group of order pq where p ≡ 1( mod q). These results come from a paper by Luthar [3]. No practical examples are done of this method. The next part deals with presenting actual groups and determining the unit structure of their integral group ring. The first two, S3 and D4 are from previous authors. The first was done by Hughes and Pearson [2], the second by C. Polcino- Milies [4]. I also present, D6, which is new result of the thesis.
Proceedings of the American Mathematical Society
On a conjecture of Zassenhaus on torsion units in integral group rings. II1986 •
2021 •
For a finite group G and U: = U(ℤG), the group of units of the integral group ring of G, we study the implications of the structure of G on the abelianization U/U' of U. We pose questions on the connections between the exponent of G/G' and the exponent of U/U' as well as between the ranks of the torsion-free parts of Z(U), the center of U, and U/U'. We show that the units originating from known generic constructions of units in ℤG are well-behaved under the projection from U to U/U' and that our questions have a positive answer for many examples. We then exhibit an explicit example which shows that the general statement on the torsion-free part does not hold, which also answers questions from [BJJ^+18].
Israel Journal of Mathematics
On the normalizer problem for integral group rings of torsion groups2002 •
2023 •
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