Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-15T07:00:09.099Z Has data issue: false hasContentIssue false
  • English
  • Français

On Class Sums in p-Adic Group Rings

Published online by Cambridge University Press:  20 November 2018

Sudarshan K. Sehgal
Sudarshan K. Sehgal*
Affiliation:
University of Alberta, Edmonton, Alberta
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.

In this note we prove that an isomorphism of p-adic group rings of finite p-groups maps class sums onto class sums. For integral group rings this is a well known theorem of Glauberman (see [3; 7]). As an application, we show that any automorphism of the p-adic group ring of a finite p-group of nilpotency class 2 is composed of a group automorphism and a conjugation by a suitable element of the p-adic group algebra. This was proved for integral group rings of finite nilpotent groups of class 2 in [5]. In general this question remains open. We also indicate an extension of a theorem of Passman and Whitcomb. The following notation is used.

G denotes a finite p-group.

Z denotes the ring of (rational) integers.

ZP denotes the ring of p-adic integers.

Qp denotes the p-adic number field.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 3 , June 1971 , pp. 541 - 543
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras (Interscience, New York, 1962).Google Scholar
2. Hasse, H., Zahlentheorie (Académie-Verlag, Berlin, 1963).Google Scholar
3. Passman, D. S., Isomorphic groups and group rings, Pacific J. Math. 15 (1965), 561–583.Google Scholar
4. Sandling, R., Note on the integral group ring problem (to appear).Google Scholar
5. Sehgal, S. K., On the isomorphism of integral group rings. I, Can. J. Math. 21 (1969), 410–413.Google Scholar
6. Sehgal, S. K., On the isomorphism of p-adic group rings, J. Number Theory 2 (1970), 500–508.Google Scholar
7. Whitcomb, A., The group ring problem, Ph.D. Thesis, University of Chicago, Illinois, 1968.Google Scholar