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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Main Author:	Ilangovan, Sheila
Format:	Thesis
Language:	English
Published:	2013
Subjects:	QA Mathematics
Online Access:	http://eprints.utm.my/id/eprint/43966/5/SheilaIlangovanPFS2013.pdf
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id	my-utm-ep.43966
record_format	uketd_dc
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

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

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