Canonical bases for normal subgroups of finitely generated free groups with finite abelian factor groups

Canonical bases for normal subgroups of finitely generated free groups with finite abelian factor... Abstract. Free bases for normal subgroups of finitely generated free groups with finite abelian factor groups are constructed. These bases consist of powers of the generators and simple basic commutators in the generators of the free group. Applications concerning /?-dimension subgroups (p a prime) and the construction of irreducible modules of linear groups are discussed. 1991 Mathematics Subject Classification: 20E07, 20E05. 0. Introduction Let 3F be a free group of rank d and let ^ be a subgroup of finite index n\ it is well known that ^ is then a free group on l 4- n(d-- 1) generators. Furthermore, both Nielsen's and Schreier's methods (see for instance [Ha]) to show this result are constructive: they exhibit an explicit free basis for ^. However, we notice that in some applications (e.g. see Section 5) it may be useful to be able to impose some conditions on the form that the elements of such a basis will have, äs words in the generators. Our main result here (Theorem 1.3) is the construction of free bases C and D for a normal subgroup Jf < &, provided 3F\Jf is finite abelian (for a basis of the commutator subgroup of 3F http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Canonical bases for normal subgroups of finitely generated free groups with finite abelian factor groups

Forum Mathematicum, Volume 5 (5) – Jan 1, 1993

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Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.1993.5.23
Publisher site
See Article on Publisher Site

Abstract

Abstract. Free bases for normal subgroups of finitely generated free groups with finite abelian factor groups are constructed. These bases consist of powers of the generators and simple basic commutators in the generators of the free group. Applications concerning /?-dimension subgroups (p a prime) and the construction of irreducible modules of linear groups are discussed. 1991 Mathematics Subject Classification: 20E07, 20E05. 0. Introduction Let 3F be a free group of rank d and let ^ be a subgroup of finite index n\ it is well known that ^ is then a free group on l 4- n(d-- 1) generators. Furthermore, both Nielsen's and Schreier's methods (see for instance [Ha]) to show this result are constructive: they exhibit an explicit free basis for ^. However, we notice that in some applications (e.g. see Section 5) it may be useful to be able to impose some conditions on the form that the elements of such a basis will have, äs words in the generators. Our main result here (Theorem 1.3) is the construction of free bases C and D for a normal subgroup Jf < &, provided 3F\Jf is finite abelian (for a basis of the commutator subgroup of 3F

Journal

Forum Mathematicumde Gruyter

Published: Jan 1, 1993

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