Nilpotent Q[x] -powered groups and Z[x] -groups.
Item
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Title
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Nilpotent Q[x] -powered groups and Z[x] -groups.
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Identifier
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AAI3127896
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identifier
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3127896
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Creator
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Majewicz, Stephen.
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Contributor
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Adviser: Gilbert Baumslag
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Date
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2004
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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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An exponential group or A-group (as defined by A. G. Myasnikov and V. N. Remeslennikov [12]) is a group, G, equipped with an action by an associative ring with unity, A, such that for all g ∈ G and for all alpha ∈ A, the element g alpha ∈ G is uniquely defined and the following axioms hold: (1) g1 = g, galphag beta = galpha+beta, (galpha)beta = galphabeta for all g ∈ G and for all alpha, beta ∈ A. (2) ( h-1gh)alpha = h-1galpha h for all g, h ∈ G and for all alpha ∈ A. (3) If g, h ∈ G satisfy the relation [g, h ] = 1, then (gh)mu = g muhmu for all mu ∈ A.;A particular example of an exponential group is a nilpotent R-powered group, where R is a binomial ring (that is, a commutative integral domain of characteristic zero with identity such that for any r ∈ R and k ∈ Z+ , ( rk ) ∈ R, where ( rk ) = rr-1&cdots; r-k+1k! . A nilpotent R-powered group (see P. Hall [5], [6] and R. B. Warfield, Jr. [14]) is a nilpotent group, G, equipped with an action by R such that, for all g ∈ G and for all alpha ∈ R, the element galpha ∈ G is uniquely defined and the following axioms hold: (1) g1 = g, galphagbeta = galpha+beta, (g alpha)beta = galphabeta for all g ∈ G and for all alpha, beta ∈ R. (2) (h-1gh )alpha = h-1 galphah for all g, h ∈ G and for all alpha ∈ R. (3) ga1&cdots;ga n = tau1(g¯)alphatau 2 g&d1; a2 ···tauk-1 g&d1; ak-1 tauk g&d1; ak for gi ∈ G and for every alpha ∈ R, where taui( g¯) = taui(g 1,...,gn) and k is the class of gp(g1,... gn). The taui(g¯ )'s are known as the Hall-Petresco words.;In this thesis I generalize the notion of a nilpotent group in two specific classes of exponential groups, namely the class of nilpotent Q [x]-powered groups and the class of Z [x]-groups. Many questions which arise in the study of ordinary nilpotent groups are explored in this thesis for these particular exponential groups. I also introduce several new concepts and results for nilpotent Q [x]-powered groups and Z [x]-groups in general, expanding on the existing theory. Some known results for nilpotent R-powered groups, where R is any binomial ring, are mentioned in the papers of P. Hall ([5], [6]) and in the book by R. B. Warfield, Jr. [14] (A. M. Duguid also has results for such groups, but I was unable to find any of these results in the literature). R. C. Lyndon [10] and A. G. Myasnikov and V. N. Remeslennikov [12] have studied free A-groups, where A is any associative ring with unity. In [12], some foundational work for A-groups is given as well.
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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.
