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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 http://eprints.utm.my/id/eprint/43966/
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id	my.utm.43966
record_format	eprints
spelling	my.utm.439662017-06-22T02:28:43Z http://eprints.utm.my/id/eprint/43966/ Conjugacy classes and graphs of two-groups of nilpotency class two 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 NonPeerReviewed application/pdf en http://eprints.utm.my/id/eprint/43966/5/SheilaIlangovanPFS2013.pdf Ilangovan, Sheila (2013) Conjugacy classes and graphs of two-groups of nilpotency class two. PhD thesis, Universiti Teknologi Malaysia, Faculty of Science.
institution	Universiti Teknologi Malaysia
building	UTM Library
collection	Institutional Repository
continent	Asia
country	Malaysia
content_provider	Universiti Teknologi Malaysia
content_source	UTM Institutional Repository
url_provider	http://eprints.utm.my/
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
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
publishDate	2013
url	http://eprints.utm.my/id/eprint/43966/5/SheilaIlangovanPFS2013.pdf http://eprints.utm.my/id/eprint/43966/
_version_	1643651330291531776
score	13.211869

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

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