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A Note on Nielsen Reduction and Coset Enumeration

  • Groups can be studied using methods from different fields such as combinatorial group theory or string rewriting. Recently techniques from Gröbner basis theory for free monoid rings (non-commutative polynomial rings) respectively free group rings have been added to the set of methods due to the fact that monoid and group presentations (in terms of string rewriting systems) can be linked to special polynomials called binomials. In the same mood, the aim of this paper is to discuss the relation between Nielsen reduced sets of generators and the Todd-Coxeter coset enumeration procedure on the one side and the Gröbner basis theory for free group rings on the other. While it is well-known that there is a strong relationship between Buchberger's algorithm and the Knuth-Bendix completion procedure, and there are interpretations of the Todd-Coxeter coset enumeration procedure using the Knuth-Bendix procedure for special cases, our aim is to show how a verbatim interpretation of the Todd-Coxeter procedure can be obtained by linking recent Gröbner techniques like prefix Gröbner bases and the FGLM algorithm as a tool to study the duality of ideals. As a side product our procedure computes Nielsen reduced generating sets for subgroups in finitely generated free groups.

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Author:Birgit Reinert, Klaus Madlener
URN (permanent link):urn:nbn:de:hbz:386-kluedo-4416
Serie (Series number):Reports on Computer Algebra (ZCA Report) (19)
Document Type:Preprint
Language of publication:English
Year of Completion:1998
Year of Publication:1998
Publishing Institute:Technische Universität Kaiserslautern
Date of the Publication (Server):2000/04/03
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
Licence (German):Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011