Transactions of the American Mathematical Society

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ISSN 1088-6850(e) ISSN 0002-9947(p)

Arithmetic rigidity and units in group rings

Author(s): F. E. A. Johnson
Journal: Trans. Amer. Math. Soc. 353 (2001), 4623-4635.
MSC (2000): Primary 20C05; Secondary 20G20, 22E40
Posted: May 9, 2001
MathSciNet review: 1851185
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Abstract:

For any finite group the group $U(\mathbf{Z}[G])$ of units in the integral group ring $\mathbf{Z}[G]$ is an arithmetic group in a reductive algebraic group, namely the Zariski closure of $\mathbf{SL}_1(\mathbf{Q}[G])$ . In particular, the isomorphism type of the $\mathbf{Q}$ -algebra $\mathbf{Q}[G]$ determines the commensurability class of $U(\mathbf{Z}[G])$ ; we show that, to a large extent, the converse is true. In fact, subject to a certain restriction on the $\mathbf{Q}$ -representations of the converse is exactly true.

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