Ngcibi, Sakhile L. (2006) Studies of equivalent fuzzy subgroups of finite abelian pGroups of rank two and their subgroup lattices. PhD thesis, Rhodes University.

Text
Thesis_SakhileNgcibi.pdf 1556Kb 
Abstract
We determine the number and nature of distinct equivalence classes of fuzzy subgroups of finite Abelian pgroup G of rank two under a natural equivalence relation on fuzzy subgroups. Our discussions embrace the necessary theory from groups with special emphasis on finite pgroups as a step towards the classification of crisp subgroups as well as maximal chains of subgroups. Unique naming of subgroup generators as discussed in this work facilitates counting of subgroups and chains of subgroups from subgroup lattices of the groups. We cover aspects of fuzzy theory including fuzzy (homo) isomorphism together with operations on fuzzy subgroups. The equivalence characterization as discussed here is finer than isomorphism. We introduce the theory of keychains with a view towards the enumeration of maximal chains as well as fuzzy subgroups under the equivalence relation mentioned above. We discuss a strategy to develop subgroup lattices of the groups used in the discussion, and give examples for specific cases of prime p and positive integers n,m. We derive formulas for both the number of maximal chains as well as the number of distinct equivalence classes of fuzzy subgroups. The results are in the form of polynomials in p (known in the literature as Hall polynomials) with combinatorial coefficients. Finally we give a brief investigation of the results from a graphtheoretic point of view. We view the subgroup lattices of these groups as simple, connected, symmetric graphs.
Item Type:  Thesis (PhD) 

Additional Information:  Ph.D. (Mathematics) 
Uncontrolled Keywords:  Fuzzy subgroups, Rank two, Equivalence relation, Maximal chains, Keychains, Invariants, Solvability, Isomorphic subgroups, Equivalence classes, Lattice diagrams, Partition, Tree orientations, Connectivity, Paths 
Subjects:  Y Unknown > Subjects to be assigned 
Divisions:  Faculty > Faculty of Science > Mathematics (Pure & Applied) 
Supervisors:  Murali, V. (Prof.) 
ID Code:  1307 
Deposited By:  Rhodes Library Archive Administrator 
Deposited On:  12 Mar 2009 
Last Modified:  01 Aug 2012 10:15 
316 fulltext download(s) in the past 12 months
More statistics...
Repository Staff Only: item control page