Skip to content

Research at St Andrews

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

Research output: Contribution to journalArticle


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).


Original languageEnglish
Pages (from-to)263-283
JournalJournal of K-theory
Issue number2
Early online date16 Feb 2010
Publication statusPublished - Oct 2010

Discover related content
Find related publications, people, projects and more using interactive charts.

View graph of relations

Related by author

  1. YangBaxter - Combinatorial Solutions for the Yang-Baxter equation, Version 0.7.0 (GAP package)

    Vendramin, L. & Konovalov, A., 23 Feb 2019

    Research output: Non-textual formSoftware

  2. GAP – Groups, Algorithms, and Programming, Version 4.10.1

    The GAP Group, Behrends, R., Breuer, T., Horn, M., Hulpke, A., Jefferson, C. A., Konovalov, A., Linton, S. A., Lübeck, F., Mitchell, J. D., Pfeiffer, M. J., Siccha, S. & Torpey, M. C., 23 Feb 2019

    Research output: Non-textual formSoftware

  3. On skew braces and their ideals

    Konovalov, A., Smoktunowicz, A. & Vendramin, L., 22 Dec 2018, In : Experimental Mathematics. Latest Articles, 10 p.

    Research output: Contribution to journalArticle

  4. LAGUNA - Lie AlGebras and UNits of group Algebras, Version 3.9.1 (Refereed GAP package)

    Bovdi, V., Konovalov, A., Rossmanith, R. & Schneider, C., 30 Nov 2018

    Research output: Non-textual formSoftware

ID: 4847670