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Title: Algorithms for subgroup presentations : computer implementation and applications
Author: Heggie, Patricia M.
Awarding Body: University of St Andrews
Current Institution: University of St Andrews
Date of Award: 1991
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One of the main algorithms of computational group theory is the Todd-Coxeter coset enumeration algorithm, which provides a systematic method for finding the index of a subgroup of a finitely presented group. This has been extended in various ways to provide not only the index of a subgroup, but also a presentation for the subgroup. These methods tie in with a technique introduced by Reidemeister in the 1920's and later improved by Schreier, now known as the Reidemeister-Schreier algorithm. In this thesis we discuss some of these variants of the Todd-Coxeter algorithm and their inter-relation, and also look at existing computer implementations of these different techniques. We then go on to describe a new package for coset methods which incorporates various types of coset enumeration, including modified Todd- Coxeter methods and the Reidemeister-Schreier process. This also has the capability of carrying out Tietze transformation simplification. Statistics obtained from running the new package on a collection of test examples are given, and the various techniques compared. Finally, we use these algorithms, both theoretically and as computer implementations, to investigate a particular class of finitely presented groups defined by the presentation: < a, b | an = b2 = (ab-1) ß =1, ab2 = ba2 >. Some interesting results have been discovered about these groups for various values of β and n. For example, if n is odd, the groups turn out to be finite and metabelian, and if β= 3 or β= 4 the derived group has an order which is dependent on the values of n (mod 8) and n (mod 12) respectively.
Supervisor: Robertson, E. F. Sponsor: Science and Engineering Research Council (SERC)
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA171.C7H3 ; Group theory