2015
DOI: 10.1016/j.jpaa.2014.09.031
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Describing units of integral group rings up to commensurability

Abstract: We restrict the types of 2 × 2-matrix rings which can occur as simple components in the Wedderburn decomposition of the rational group algebra of a finite group. This results in a description up to commensurability of the group of units of the integral group ring ZG for all finite groups G that do not have a non-commutative Frobenius complement as a quotient.

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Cited by 14 publications
(23 citation statements)
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References 24 publications
(51 reference statements)
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“…We hence embed SL 2 (( x,y Z )) into SL 2 (( −1,−1 R )) and re-use the same technique. Note that in [10], the authors showed that the exceptional components of type 3 can be reduced to the following three matrix rings over quaternion algebras:…”
Section: Introductionmentioning
confidence: 99%
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“…We hence embed SL 2 (( x,y Z )) into SL 2 (( −1,−1 R )) and re-use the same technique. Note that in [10], the authors showed that the exceptional components of type 3 can be reduced to the following three matrix rings over quaternion algebras:…”
Section: Introductionmentioning
confidence: 99%
“…This makes it possible to describe generators of SL 2 (O) with O a maximal order in these algebras by imitating the classical way, based on the Euclidean algorithm, in which generators for SL 2 (Z) are computed. For details see [10,Proposition 4.1]. So for the pure purpose of units in group rings, there is a way of computing generators for the unit group of a maximal order in components of type 3 and the details may be found in [10].…”
Section: Introductionmentioning
confidence: 99%
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“…Q ) are very limited. It was proven by Eisele, Kiefer and Van Gelder [14] that only a finite number of division algebras can occur and, moreover, the possible parameters have been described. Together with the results of Bächle, Janssens, Jespers, Kiefer and D. Temmerman in [2] one has the following result.…”
Section: Rational Wedderburn Decompositionmentioning
confidence: 99%
“…For producing this table, the classifications of either Banieqbal or Nebe can be used. The finite subgroups having M 2 (Q) or M 2 (K) with K a quadratic imaginary extension of Q are shown to be a subgroup of GL(2, 25) in [EKVG15].…”
Section: Introductionmentioning
confidence: 99%