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The Group of Units of the Integral Group Ring ZS3

Published online by Cambridge University Press:  20 November 2018

I. Hughes  and
K. R. Pearson
I. Hughes
Affiliation:
Queen's University, Kingston, Ontario
K. R. Pearson
Affiliation:
Pennsylvania State University, University Park, Pennsylvania
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We denote by ZG the integral group ring of the finite group G. We call ±g, for g in G, a trivial unit of ZG. For G abelian, Higman [4] (see also [3, p. 262 ff]) showed that every unit of finite order in ZG is trivial. For arbitrary finite G (indeed, for a torsion group G, not necessarily finite), Higman [4] showed that every unit in ZG is trivial if and only if G is

  1. (i) abelian and the order of each element divides 4, or

  2. (ii) abelian and the order of each element divides 6, or

  3. (iii) the direct product of the quaternion group of order 8 and an abelian group of exponent 2.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 4 , December 1972 , pp. 529 - 534
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Berman, S. D., On the equation xm = l in an integral group ring, Ukrain. Mat. Ž. 7 (1955), 253-261.Google Scholar
2. Boerner, H., Representations of groups, North-Holland, Amsterdam, 1963.Google Scholar
3. Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras, Interscience, New York, 1962.Google Scholar
4. Higman, G., The units of group rings, Proc. London Math. Soc. 46 (1940), 231-248.Google Scholar
5. Kurosh, A., The theory of groups, Vol. 2, Chelsea, New York, 1956.Google Scholar
6. Sehgal, S. K., On the isomorphism of integral group rings I, Canadian J. Math. 21 (1969), 410-413.Google Scholar
7. Takahashi, M., Bemerkungen über den Untergruppensatz in freien Produkten, Proc. Japan Acad. 20 (1944), 589-594.Google Scholar
8. Takahashi, M., Some properties of the group ring over rational integers of a finite group, Notices Amer. Math. Soc. 12 (1965), p. 463.Google Scholar
9. Taussky, O., Matrices of rational integers, Bull. Amer. Math. Soc. 66 (1960), 327-345.Google Scholar