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

Published online by Cambridge University Press:  16 February 2010

Ann Dooms ,
Eric Jespers  and
Alexander Konovalov
Ann Dooms
Affiliation:
Vrije Universiteit Brussel, Interdisciplinary Institute for Broadband Technology (IBBT), Department of Electronics and Informatics (ETRO), Pleinlaan 2, B-1050 Brussels, Belgium, andooms@vub.ac.be
Eric Jespers
Affiliation:
Department of Mathematics, Faculty of Sciences, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium, efjesper@vub.ac.be
Alexander Konovalov
Affiliation:
School of Computer Science, University of St Andrews, Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland, alexk@mcs.st-andrews.ac.uk
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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 M2(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 PSL2(ℤ). Furthermore, for each simple Wedderburn component M2(ℚ) of ℚG, the new generators give a free subgroup that is embedded in M2(ℤ).

Keywords

Farey unitcongruence subgroupintegral group ring
Type
Research Article
Information
Journal of K-Theory , Volume 6 , Issue 2 , October 2010 , pp. 263 - 283
Copyright
Copyright © ISOPP 2010

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