Groups of units of integral group rings of Kleinian type

Pita, Antonio; del Río, Ángel; Ruiz, Manuel

doi:10.1090/S0002-9947-04-03574-3

Groups of units of integral group rings of Kleinian type
HTML articles powered by AMS MathViewer

by Antonio Pita, Ángel del Río and Manuel Ruiz

Trans. Amer. Math. Soc. 357 (2005), 3215-3237

DOI: https://doi.org/10.1090/S0002-9947-04-03574-3

Published electronically: October 7, 2004

PDF | Request permission

Abstract:

We explore a method to obtain presentations of the group of units of an integral group ring of some finite groups by using methods on Kleinian groups. We classify the nilpotent finite groups with central commutator for which the method works and apply the method for two concrete groups of order 16.

References

Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777, DOI 10.1007/978-1-4612-1146-4
L. Bianchi, Sui gruppi de sostituzioni lineari con coeficienti appartenente a corpi quadratici imaginari, Math. Ann. 40 (1892), 332–412.
J. Elstrodt, F. Grunewald, and J. Mennicke, Groups acting on hyperbolic space, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. Harmonic analysis and number theory. MR 1483315, DOI 10.1007/978-3-662-03626-6
Benjamin Fine, Algebraic theory of the Bianchi groups, Monographs and Textbooks in Pure and Applied Mathematics, vol. 129, Marcel Dekker, Inc., New York, 1989. MR 1010229
L.R. Ford, Automorphic Functions (second edition), Chelsea, New York, 1951.
Roderick Gow and Bertram Huppert, Degree problems of representation theory over arbitrary fields of characteristic $0$. On theorems of N. Itô and J. G. Thompson, J. Reine Angew. Math. 381 (1987), 136–147. MR 918845, DOI 10.1515/crll.1987.381.136
Eric Jespers and Guilherme Leal, Describing units of integral group rings of some $2$-groups, Comm. Algebra 19 (1991), no. 6, 1809–1827. MR 1113959, DOI 10.1080/00927879108824230
E. Jespers and G. Leal, Degree $1$ and $2$ representations of nilpotent groups and applications to units of group rings, Manuscripta Math. 86 (1995), no. 4, 479–498. MR 1324684, DOI 10.1007/BF02568007
Eric Jespers, Guilherme Leal, and Angel del Río, Products of free groups in the unit group of integral group rings, J. Algebra 180 (1996), no. 1, 22–40. MR 1375564, DOI 10.1006/jabr.1996.0050
E. Jespers and M. M. Parmenter, Units of group rings of groups of order $16$, Glasgow Math. J. 35 (1993), no. 3, 367–379. MR 1240380, DOI 10.1017/S0017089500009952
Guilherme Leal and C. Polcino Milies, Isomorphic group (and loop) algebras, J. Algebra 155 (1993), no. 1, 195–210. MR 1206630, DOI 10.1006/jabr.1993.1039
Bernard Maskit, Kleinian groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR 959135
M. M. Parmenter, Free torsion–free normal complements in integral group rings, Comm. Algebra 21 (1993), no. 10, 3611–3617. MR 1231619, DOI 10.1080/00927879308824751
Vladimir Platonov and Andrei Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press, Inc., Boston, MA, 1994. Translated from the 1991 Russian original by Rachel Rowen. MR 1278263
H. Poincaré , Mémoire sur les groupes kleinées, Acta Math. 3 (1883), 49–92.
Ángel del Río and Manuel Ruiz, Computing large direct products of free groups in integral group rings, Comm. Algebra 30 (2002), no. 4, 1751–1767. MR 1894041, DOI 10.1081/AGB-120013213
S. K. Sehgal, Units in integral group rings, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 69, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. With an appendix by Al Weiss. MR 1242557
Marie-France Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980 (French). MR 580949, DOI 10.1007/BFb0091027