Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-23T05:09:24.487Z Has data issue: false hasContentIssue false
  • English
  • Français

Outer automorphisms and non-trivial Picard groups

Published online by Cambridge University Press:  26 February 2010

M. E. Keating
M. E. Keating
Affiliation:
Imperial College, London
Get access

Extract

The Picard group P(ZG) of the integral group ring ZG is defined as the class group of two-sided invertible ZG-ideals of QG modulo those principal ideals generated by an invertible central element. The basic properties of Picard groups have been established by A. Fröhlich, I. Reiner and S. Ullom [1], [2], [3]. In this note we settle an outstanding question by exhibiting a class of finite p-groups G whose Picard groups contain nontrivial elements which are represented by principal ideals; these elements remain nontrivial in P(ZpG) also. We obtain these ideals from outer automorphisms of the groups.

Type
Research Article
Information
Mathematika , Volume 26 , Issue 1 , June 1979 , pp. 103 - 105
Copyright
Copyright © University College London 1979

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Fröhlich, A.. “The Picard group of non-commutative rings, in particular of orders” Trans. Amer. Math. Soc, 180, (1973), 1–45.CrossRefGoogle Scholar
2.Fröhlich, A., Reiner, I. and Ullom, S.. “Picard groups and class groups of orders” Proc. London Math. Soc.(3), 29 (1974) 405–434.CrossRefGoogle Scholar
3.Reiner, I.Class groups and Picard groups of group rings and orders. Regional Conference Series in Math. no. 26 (Amer. Math. Soc. 1976).CrossRefGoogle Scholar