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Search: MSC category 16S34 ( Group rings [See also 20C05, 20C07], Laurent polynomial rings )

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1. CJM 1998 (vol 50 pp. 312)

Dokuchaev, Michael A.; Singer, Maria Lucia Sobral
Units in group rings of free products of prime cyclic groups
Let $G$ be a free product of cyclic groups of prime order. The structure of the unit group ${\cal U}(\Q G)$ of the rational group ring $\Q G$ is given in terms of free products and amalgamated free products of groups. As an application, all finite subgroups of ${\cal U}(\Q G)$, up to conjugacy, are described and the Zassenhaus Conjecture for finite subgroups in $\Z G$ is proved. A strong version of the Tits Alternative for ${\cal U}(\Q G)$ is obtained as a corollary of the structural result.

Keywords:Free Products, Units in group rings, Zassenhaus Conjecture
Categories:20C07, 16S34, 16U60, 20E06

2. CJM 1997 (vol 49 pp. 1265)

Snaith, V. P.
Hecke algebras and class-group invariant
Let $G$ be a finite group. To a set of subgroups of order two we associate a $\mod 2$ Hecke algebra and construct a homomorphism, $\psi$, from its units to the class-group of ${\bf Z}[G]$. We show that this homomorphism takes values in the subgroup, $D({\bf Z}[G])$. Alternative constructions of Chinburg invariants arising from the Galois module structure of higher-dimensional algebraic $K$-groups of rings of algebraic integers often differ by elements in the image of $\psi$. As an application we show that two such constructions coincide.

Categories:16S34, 19A99, 11R65

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