Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-12-04T09:13:18.768Z Has data issue: false hasContentIssue false
  • English
  • Français

On Groups of Order p3

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The simplest example of two non-isomorphic groups with the same character tables is provided by the non-abelian groups of order p3, p ≠ 2. Let G1 be the one of exponent p and let G2 be the other. If Q denotes the field of rational numbers, then Berman (2) has shown that QG1 ≈ QG2, where QGi denotes the rational group algebra. In this note we shall show that the corresponding statement is false for ZGi where Z is the ring of rational integers. More explicitly we shall show that ZG1 does not contain a unit of order p2 so that it is impossible to embed ZG2 in ZG1.

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