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From Farey symbols to generators for subgroups of finite index in integral group rings of finite groups

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Abstract

The topic of this paper is the construction of a finite set of generators for a subgroup of finite index in the unit group U(ZG) of the integral group ring of a finite group G. The present paper is a continuation of earlier research by Bass and Milnor, Jespers and Leal, and Ritter and Sehgal who constructed such generators provided that the group G does not have a non-abelian fixed-point free epimorphic image and the rational group algebra QG does not have simple epimorphic images that are two-by-two matrices over either the rationals, a quadratic imaginary extension of the rationals or a non-commutative division algebra. In this paper we allow simple images of the type M(2)(Q). We will do so by introducing new additional generators using Farey symbols, which are in one to one correspondence with fundamental polygons of congruence subgroups of PSL(2)(Z). Furthermore, for each simple Wedderburn component M(2)(Q) of QG, the new generators give a free subgroup that is embedded in M(2)(Z).
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Original languageEnglish
Pages (from-to)263-283
JournalJournal of K-theory
Volume6
Issue number2
Early online date16 Feb 2010
DOIs
StatePublished - Oct 2010

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