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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
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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

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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