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On Groups of Order p 3

Published online by Cambridge University Press:  20 November 2018

James A. Cohn  and
Donald Livingstone
James A. Cohn
Affiliation:
University of Michigan, Ann Arbor, Michigan
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The simplest example of two non-isomorphic groups with the same character tables is provided by the non-abelian groups of order p 3, p ≠ 2. Let G 1 be the one of exponent p and let G 2 be the other. If Q denotes the field of rational numbers, then Berman (2) has shown that QG 1 ≈ QG 2, where QG i denotes the rational group algebra. In this note we shall show that the corresponding statement is false for ZG i where Z is the ring of rational integers. More explicitly we shall show that ZG 1 does not contain a unit of order p 2 so that it is impossible to embed ZG 2 in ZG 1.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 15 , 1963 , pp. 622 - 624
Copyright
Copyright © Canadian Mathematical Society 1963

References

1. Berman, S. D., On certain properties of integral group rings, Dokl. Akad. Nauk SSSR(N.S.), 91 (1953), 7–9; M.R. 15, 99.Google Scholar
2. Berman, S. D., On certain properties of group rings over the field of rational numbers, Uzgorod. Gos. Univ. Naucn. Zap. Him. Fiz. Mat., 12 (1955), 88–110; M.R. 20 ,No. 3920. (This article was not available to us, and so our knowledge of it is based upon the review.)Google Scholar