Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-11T03:27:52.030Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Banieqbal, B., Classification of finite subgroups of 2×2 matrices over a division algebra of characteristic zero, J. Algebra 119 (1988), 449–512.CrossRefGoogle Scholar
2.Borel, A.A. and Chandra, Harish, Arithmetic Subgroups Of Algebraic Groups, Ann. of Math. 75 (1962), 485–535.CrossRefGoogle Scholar
3.Bovdi, V., Konovalov, A., Rossmanith, R. and Schneider, C., LAGUNA — Lie AlGebras and UNits of group Algebras, Version 3.4, GAP package, 2007 (http://www.cs.st-andrews.ac.uk/~alexk/laguna.htm).Google Scholar
4.Cristo, O. Broche, Konovalov, A., Olivieri, A., Olteanu, G. and del Río, Á., Wedderga — Wedderburn Decomposition of Group Algebras, Version 4.3.2, GAP package, 2008 (http://www.um.es/adelrio/wedderga.htm).Google Scholar
5.Chan, S., Lang, M., Lim, C. and Tan, S.P., Special polygons for subgroups of the modular group and applications, Internat. J. Math. 4 (1993), no. 1, 11–34.CrossRefGoogle Scholar
6.Dooms, A., Unitary units in integral group rings, J. Algebra Appl. 5 (2006), no. 1, 43–52.CrossRefGoogle Scholar
7.Dooms, A., Jespers, E., Konovalov, A. and Verril, H.. Congruence — Congruence subgroups of SL2(ℤ), GAP package, in development.Google Scholar
8.Dooms, A., Jespers, E. and Ruiz, M., Free groups and subgroups of finite index in the unit group of an integral group ring, Comm. Algebra 35 (2007), no. 9, 2879–2888.CrossRefGoogle Scholar
9.The GAP Group, GAP — Groups, Algorithms, and Programming, Version 4.4.10; 2007, (http://www.gap-system.org).Google Scholar
10.Gonçalves, J.Z. and del Río, Á., Bicyclic units, Bass cyclic units and free groups, J. Group Theory 11 (2008), 247–265.CrossRefGoogle Scholar
11.Gonçalves, J.Z. and Passman, D.S., Embedding free products in the unit group of an integral group ring, Arch. Math. (Basel) 82 (2004), no. 2, 97–102. 16S34CrossRefGoogle Scholar
12.Gonçalves, J.Z. and Passman, D.S., Linear groups and group rings, J. Algebra 295 (2006), no. 1, 94–118.CrossRefGoogle Scholar
13.Gonçalves, J.Z. and Passman, D.S., Erratum to: “Linear groups and group rings, J. Algebra 307 (2007), no. 2, 930–931.CrossRefGoogle Scholar
14.Gonçalves, J.Z. and Passman, D.S., Involutions and free pairs of bicyclic units in integral group rings, preprint.Google Scholar
15.Hartley, B. and Pickel, P.F., Free Subgroups in the Unit Groups of Integral Group Rings, Canadian J. Math. 32 (1980), 1342–1352.CrossRefGoogle Scholar
16.Jespers, E. and Leal, G., Describing units of integral group rings of some 2-groups, Comm. Algebra 19 (1991), no. 6, 1809–1827.CrossRefGoogle Scholar
17.Jespers, E. and Leal, G., Generators of large subgroups of the unit group of integral group rings, Manuscripta Math. 78 (1993), no. 3, 303–315.CrossRefGoogle Scholar
18.Jespers, E. and Leal, G., Degree 1 and 2 representations of nilpotent groups and applications to units of group rings, Manuscripta Math. 86 (1995), no. 4, 479–498.CrossRefGoogle Scholar
19.Jespers, E. and Parmenter, M.M., Bicyclic units in ZS3, Bull. Soc. Math. Belg. Sér. B 44 (1992), no. 2, 141–146.Google Scholar
20.Jespers, E., Parmenter, M.M. and Sehgal, S.K., Central Units Of Integral Group Rings Of Nilpotent Groups, Proc. Amer. Math. Soc. 124 (1996), no. 4, 1007–1012.CrossRefGoogle Scholar
21.Jespers, E., del Río, Á. and Ruiz, M., Groups generated by two bicyclic units in integral group rings, J. Group Theory 5 (2002), no. 4, 493–511.CrossRefGoogle Scholar
22.Kleinert, E., A theorem on units of integral group rings, J. Pure and Applied Algebra 49 (1987), 161–171.CrossRefGoogle Scholar
23.Kulkarni, R.S., An arithmetic-geometric method in the study of the subgroups of the modular group, Amer. J. Math. 113 (1991), no. 6, 1053–1133.CrossRefGoogle Scholar
24.Lang, M., Lim, C. and Tan, S.P., An algorithm for determining if a subgroup of the modular group is congruence, J. London Math. Soc. (2) 51 (1995), no. 3, 491–502.CrossRefGoogle Scholar
25.Marciniak, Z. and Sehgal, S.K., Constructing free subgroups of integral group ring units, Proc. Amer. Math. Soc. 125 (1997), no. 4, 1005–1009.CrossRefGoogle Scholar
26.Maskit, B., Kleinian groups, Grundlehren der mathematischen Wissenschaften 287, Springer-Verlag, Berlin, 1980.Google Scholar
27.Newman, M., Integral matrices, Pure and Applied Mathematics 45, Academic Press, New York-London, 1972.Google Scholar
28.Ritter, J. and Sehgal, S.K., Certain normal subgroups of units in group rings, J. Reine Angew. Math. 381 (1987), 214–220.Google Scholar
29.Ritter, J. and Sehgal, S.K., Generators of subgroups of ***u(ZG), Representation theory, group rings, and coding theory, 331–347, Contemp. Math. 93, Amer.Math. Soc., Providence, RI, 1989.Google Scholar
30.Ritter, J. and Sehgal, S.K., Construction of units in integral group rings of finite nilpotent groups, Bull. Amer. Math. Soc. 20 (1989), 165–168.CrossRefGoogle Scholar
31.Ritter, J. and Sehgal, S.K., Construction of units in integral group rings of finite nilpotent groups, Trans. Amer.Math. Soc. 324 (2) (1991), 603–621.CrossRefGoogle Scholar
32.Ritter, J. and Sehgal, S.K., Construction of units in group rings of monomial and symmetric groups, J. Algebra 142 (1991), 511–526.CrossRefGoogle Scholar
33.Ritter, J. and Sehgal, S.K., Units of group rings of solvable and Frobenius groups over large rings of cyclotomic units, J. Algebra 158 (1) (1993), 116–129.CrossRefGoogle Scholar
34.Salwa, A., On free subgroups of units of rings, Proc. Amer. Math. Soc. 127 (1999), no. 9, 2569–2572.CrossRefGoogle Scholar
35.Sehgal, S.K., Units In Integral Group Rings, Longman Scientific and Technical Press, Harlow, 1993.Google Scholar