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Subgroups of bounded Abelian groups

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Date Issued:
2004
Summary:
Birkhoff raised the question of how to determine "relative invariants of subgroups" of a given group. Let us consider pairs (A, B ) where B is a finite pn-bounded Abelian group and A is a subgroup of B. Maps between pairs (A, B) --> (A', B') are morphisms f : B --> B' such that f (A) --> A'. Classification of such pairs, up to isomorphism, is Birkhoff's famous problem. By the Krull-Remak-Schmidt theorem, an arbitrary pair (A, B) is a direct sum of indecomposable pairs, and the multiplicities of the indecomposables are determined uniquely. The purpose of this thesis is to describe the decomposition of such pairs, (A, B), explicitly for n = 2 and n = 3. We describe explicitly how an indecomposable pair can possibly embed into a given pair (A, B). This construction gives rise to formulas for the multiplicity of an indecomposable in the direct sum decomposition of the pair (A, B). These decomposition numbers form a full set of relative invariant, as requested by Birkhoff.
Title: Subgroups of bounded Abelian groups.
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Name(s): Petroro, Carla.
Florida Atlantic University, Degree grantor
Schmidmeier, Markus, Thesis advisor
Type of Resource: text
Genre: Electronic Thesis Or Dissertation
Issuance: monographic
Date Issued: 2004
Publisher: Florida Atlantic University
Place of Publication: Boca Raton, Fla.
Physical Form: application/pdf
Extent: 68 p.
Language(s): English
Summary: Birkhoff raised the question of how to determine "relative invariants of subgroups" of a given group. Let us consider pairs (A, B ) where B is a finite pn-bounded Abelian group and A is a subgroup of B. Maps between pairs (A, B) --> (A', B') are morphisms f : B --> B' such that f (A) --> A'. Classification of such pairs, up to isomorphism, is Birkhoff's famous problem. By the Krull-Remak-Schmidt theorem, an arbitrary pair (A, B) is a direct sum of indecomposable pairs, and the multiplicities of the indecomposables are determined uniquely. The purpose of this thesis is to describe the decomposition of such pairs, (A, B), explicitly for n = 2 and n = 3. We describe explicitly how an indecomposable pair can possibly embed into a given pair (A, B). This construction gives rise to formulas for the multiplicity of an indecomposable in the direct sum decomposition of the pair (A, B). These decomposition numbers form a full set of relative invariant, as requested by Birkhoff.
Identifier: 9780496233670 (isbn), 13118 (digitool), FADT13118 (IID), fau:9981 (fedora)
Note(s): Charles E. Schmidt College of Science
Thesis (M.S.)--Florida Atlantic University, 2004.
Subject(s): Abelian groups
Modules (Algebra)
Indecomposable modules
Representations of groups
Algebras, Linear
Held by: Florida Atlantic University Libraries
Persistent Link to This Record: http://purl.flvc.org/fcla/dt/13118
Sublocation: Digital Library
Use and Reproduction: Copyright © is held by the author with permission granted to Florida Atlantic University to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Use and Reproduction: http://rightsstatements.org/vocab/InC/1.0/
Host Institution: FAU
Is Part of Series: Florida Atlantic University Digital Library Collections.