Groups in Which Commutativity Is a Transitive Relation
Wu, Yu-Fen
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https://hdl.handle.net/2142/86964
Description
Title
Groups in Which Commutativity Is a Transitive Relation
Author(s)
Wu, Yu-Fen
Issue Date
1998
Doctoral Committee Chair(s)
Derek J.S.Robinson
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
A group G is called a CT-group if commutativity is a transitive relation on the set of non-identity elements of G. This dissertation is concerned with the class of CT-groups. Finite and locally finite CT-groups are studied in detail with structure theorems given in both solvable and insolvable cases. The investigation of solvable CT-groups leads naturally to the study of fixed-point-free groups of automorphisms of abelian groups. Topics include a rank condition for the existence of fixed-point-free groups of automorphisms of abelian torsion groups, and the extensions of abelian groups by locally finite groups with fixed-point-free actions. Torsion-free solvable CT-groups and polycyclic CT-groups are also examined in detail with constructions and structure theorems. An additional topic studied is the subclass of groups in which all quotients of subgroups are CT.
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