Conjugacy classes and graphs of two-groups of nilpotency class two

Two elements a and b of a group are called conjugate if there exists an element g in the group such that gag??1 = b: The set of all conjugates in a group forms the conjugacy classes of the group. The main objective of this research is to determine the number and size of conjugacy classes for 2-gener...

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主要作者: Ilangovan, Sheila
格式: Thesis
语言:English
出版: 2013
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spelling my-utm-ep.439662017-06-22T02:28:43Z Conjugacy classes and graphs of two-groups of nilpotency class two 2013-09 Ilangovan, Sheila QA Mathematics Two elements a and b of a group are called conjugate if there exists an element g in the group such that gag??1 = b: The set of all conjugates in a group forms the conjugacy classes of the group. The main objective of this research is to determine the number and size of conjugacy classes for 2-generator 2-groups of nilpotency class two. Suppose G is a 2-generator 2-group of class two which comprises of three types, namely Type 1, Type 2 and Type 3. The general formulas for the number of conjugacy classes of G are determined by using the base group and central extension method, respectively. It is found that for each type of the group G, the number of conjugacy classes consists of two general formulas. Moreover, the conjugacy class sizes are computed based on the order of the derived subgroup. The results are then applied into graph theory. The conjugacy class graph of G is proven as a complete graph. Consequently, some properties of the graph related to conjugacy classes of the group are found. This includes the number of connected components, diameter, the number of edges and the regularity of the graph. Furthermore, the clique number and chromatic number for groups of Type 1, 2 and 3 are shown to be identical. Besides, some properties of the graph related to commuting conjugacy classes of abelian and dihedral groups are introduced. 2013-09 Thesis http://eprints.utm.my/id/eprint/43966/ http://eprints.utm.my/id/eprint/43966/5/SheilaIlangovanPFS2013.pdf application/pdf en public phd doctoral Universiti Teknologi Malaysia, Faculty of Science Faculty of Science
institution Universiti Teknologi Malaysia
collection UTM Institutional Repository
language English
topic QA Mathematics
spellingShingle QA Mathematics
Ilangovan, Sheila
Conjugacy classes and graphs of two-groups of nilpotency class two
description Two elements a and b of a group are called conjugate if there exists an element g in the group such that gag??1 = b: The set of all conjugates in a group forms the conjugacy classes of the group. The main objective of this research is to determine the number and size of conjugacy classes for 2-generator 2-groups of nilpotency class two. Suppose G is a 2-generator 2-group of class two which comprises of three types, namely Type 1, Type 2 and Type 3. The general formulas for the number of conjugacy classes of G are determined by using the base group and central extension method, respectively. It is found that for each type of the group G, the number of conjugacy classes consists of two general formulas. Moreover, the conjugacy class sizes are computed based on the order of the derived subgroup. The results are then applied into graph theory. The conjugacy class graph of G is proven as a complete graph. Consequently, some properties of the graph related to conjugacy classes of the group are found. This includes the number of connected components, diameter, the number of edges and the regularity of the graph. Furthermore, the clique number and chromatic number for groups of Type 1, 2 and 3 are shown to be identical. Besides, some properties of the graph related to commuting conjugacy classes of abelian and dihedral groups are introduced.
format Thesis
qualification_name Doctor of Philosophy (PhD.)
qualification_level Doctorate
author Ilangovan, Sheila
author_facet Ilangovan, Sheila
author_sort Ilangovan, Sheila
title Conjugacy classes and graphs of two-groups of nilpotency class two
title_short Conjugacy classes and graphs of two-groups of nilpotency class two
title_full Conjugacy classes and graphs of two-groups of nilpotency class two
title_fullStr Conjugacy classes and graphs of two-groups of nilpotency class two
title_full_unstemmed Conjugacy classes and graphs of two-groups of nilpotency class two
title_sort conjugacy classes and graphs of two-groups of nilpotency class two
granting_institution Universiti Teknologi Malaysia, Faculty of Science
granting_department Faculty of Science
publishDate 2013
url http://eprints.utm.my/id/eprint/43966/5/SheilaIlangovanPFS2013.pdf
_version_ 1747817124904042496