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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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From Farey symbols to generators for subgroups of finite index in integral group rings of finite groups. / Dooms, Ann; Jespers, Eric; Konovalov, Alexander.

In: Journal of K-theory, Vol. 6, No. 2, 10.2010, p. 263-283.

Research output: Contribution to journalArticlepeer-review

Harvard

Dooms, A, Jespers, E & Konovalov, A 2010, 'From Farey symbols to generators for subgroups of finite index in integral group rings of finite groups', Journal of K-theory, vol. 6, no. 2, pp. 263-283. https://doi.org/10.1017/is009012013jkt079

APA

Dooms, A., Jespers, E., & Konovalov, A. (2010). From Farey symbols to generators for subgroups of finite index in integral group rings of finite groups. Journal of K-theory, 6(2), 263-283. https://doi.org/10.1017/is009012013jkt079

Vancouver

Dooms A, Jespers E, Konovalov A. From Farey symbols to generators for subgroups of finite index in integral group rings of finite groups. Journal of K-theory. 2010 Oct;6(2):263-283. https://doi.org/10.1017/is009012013jkt079

Author

Dooms, Ann ; Jespers, Eric ; Konovalov, Alexander. / From Farey symbols to generators for subgroups of finite index in integral group rings of finite groups. In: Journal of K-theory. 2010 ; Vol. 6, No. 2. pp. 263-283.

Bibtex - Download

@article{303c16a85087420ba094713a3695f2ee,
title = "From Farey symbols to generators for subgroups of finite index in integral group rings of finite groups",
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). ",
author = "Ann Dooms and Eric Jespers and Alexander Konovalov",
year = "2010",
month = oct,
doi = "10.1017/is009012013jkt079",
language = "English",
volume = "6",
pages = "263--283",
journal = "Journal of K-theory",
issn = "1865-2433",
publisher = "Cambridge University Press",
number = "2",

}

RIS (suitable for import to EndNote) - Download

TY - JOUR

T1 - From Farey symbols to generators for subgroups of finite index in integral group rings of finite groups

AU - Dooms, Ann

AU - Jespers, Eric

AU - Konovalov, Alexander

PY - 2010/10

Y1 - 2010/10

N2 - 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).

AB - 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).

U2 - 10.1017/is009012013jkt079

DO - 10.1017/is009012013jkt079

M3 - Article

VL - 6

SP - 263

EP - 283

JO - Journal of K-theory

JF - Journal of K-theory

SN - 1865-2433

IS - 2

ER -

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