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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Class groups of integral group rings
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by I. Reiner and S. Ullom PDF
Trans. Amer. Math. Soc. 170 (1972), 1-30 Request permission

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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Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 170 (1972), 1-30
  • MSC: Primary 20C05
  • DOI: https://doi.org/10.1090/S0002-9947-1972-0304470-6
  • MathSciNet review: 0304470