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Subgroups of bounded Abelian groups
 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 pnbounded 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 KrullRemakSchmidt 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 pnbounded 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 KrullRemakSchmidt 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  
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Use and Reproduction:  http://rightsstatements.org/vocab/InC/1.0/  
Host Institution:  FAU  
Is Part of Series:  Florida Atlantic University Digital Library Collections. 