Experiments in group theory: Group-theoretic algorithms in one-relator groups.
Item
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Title
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Experiments in group theory: Group-theoretic algorithms in one-relator groups.
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Identifier
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AAI9618103
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identifier
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9618103
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Creator
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Shumowitz, Marvin.
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Contributor
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Adviser: Michael Anshel
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Date
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1996
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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This dissertation improves on two algorithms in the theory of one-relator groups. The first of these is the algorithm given by Abe Shenitzer to determine if we can represent a one-relator group as the free product of two of its subgroups. Using the natural measure of complexity, the number of rewritings required, this paper gives a linear algorithm as opposed to the exponential one given by Shenitzer. The second of these algorithms involves the determination of whether a one relator group has a trivial center. This expands the work done by Gilbert Baumslag and Tekla Taylor, and complements the work of Frank Levin. In looking for groups with a trivial center, one only needs to look at two-generator groups, {dollar}G=\langle a,\ b{lcub}:{rcub}\ R=l\rangle.{dollar} The Baumslag-Taylor algorithm divides the class of one-relator groups into two subclasses, those with finite and infinitely generated normal closures of b in G. What they show is that if the group is in the later subclass and the number of b's is {dollar}>{dollar}2, then it has a trivial center. In the process, efficient pattern matching algorithms were used in improving these group-theoretic algorithms. An inference from the trivial center program is that all groups with a trivial center are those in which the normal subgroup generated by one of the variables is not finitely generated.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
