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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
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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

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