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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Class groups of integral group rings

Authors: I. Reiner and S. Ullom
Journal: Trans. Amer. Math. Soc. 170 (1972), 1-30
MSC: Primary 20C05
MathSciNet review: 0304470
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Abstract: Let $ \Lambda $ be an $ R$-order in a semisimple finite dimensional $ K$-algebra, where $ K$ is an algebraic number field, and $ R$ is the ring of algebraic integers of $ K$. Denote by $ C(\Lambda )$ the reduced class group of the category of locally free left $ \Lambda $-lattices. Choose $ \Lambda = ZG$, the integral group ring of a finite group $ G$, and let $ \Lambda '$ be a maximal $ Z$-order in $ QG$ containing $ \Lambda $. There is an epimorphism $ C(\Lambda ) \to C(\Lambda ')$, given by $ M \to \Lambda '{ \otimes _\Lambda }M$, for $ M$ a locally free $ \Lambda $-lattice. Let $ D(\Lambda )$ be the kernel of this epimorphism; the groups $ D(\Lambda ),C(\Lambda )$ and $ C(\Lambda ')$ are all finite. Our main theorem is that $ D(ZG)$ is a $ p$-group whenever $ G$ is a $ p$-group. This generalizes Fröhlich's result for the case where $ G$ is an abelian $ p$-group. Our proof uses some facts about the center $ F$ of $ QG$, as well as information about reduced norms. We also calculate $ D(ZG)$ explicitly for $ G$ cyclic of order $ 2p$, dihedral of order $ 2p$, or the quaternion group. In these cases, the ring $ ZG$ can be conveniently described by a pullback diagram.

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Article copyright: © Copyright 1972 American Mathematical Society