The probability that an element of a non-abelian group fixes a set and its applications in graph theory

The commutativity degree, defined as the probability that two randomly selected elements of a group commute, plays a very important role in determining the abelianness of a group. In this research, the commutativity degree is extended by finding the probability that a group element fixes a set. This...

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Main Author:	El-Sanfaz, Mustafa Anis
Format:	Thesis
Language:	English
Published:	2016
Subjects:	QA Mathematics
Online Access:	http://eprints.utm.my/id/eprint/78794/1/MustafaAnisElPFS2016.pdf
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spelling	my-utm-ep.787942018-09-17T04:12:54Z The probability that an element of a non-abelian group fixes a set and its applications in graph theory 2016-07 El-Sanfaz, Mustafa Anis QA Mathematics The commutativity degree, defined as the probability that two randomly selected elements of a group commute, plays a very important role in determining the abelianness of a group. In this research, the commutativity degree is extended by finding the probability that a group element fixes a set. This probability is computed under two group actions on the set namely, the conjugate action and the regular action. The set under study consists of all commuting elements of order two of metacyclic 2-groups and dihedral groups of even order. The probabilities found turned out to depend on the cardinality of the set. The results which were obtained from the probability are then linked to graph theory, more precisely to orbit graph and generalized conjugacy class graph. It is found that the orbit graph and the generalized conjugacy class graph consist of complete graphs, empty graphs or null graphs. Moreover, some graph properties including the chromatic number, clique number, dominating number and independent number are found. In addition, the necessary condition for the orbit graph and generalized conjugacy class graph to be a null graph is examined. Furthermore, two new graphs are introduced, namely the generalized commuting graph and the generalized non-commuting graph. The generalized commuting graph of all groups in the scope of this research turns out to be a union of complete graphs or null graphs, while the generalized non-commuting graph consists of regular graphs, empty graphs or null graphs. 2016-07 Thesis http://eprints.utm.my/id/eprint/78794/ http://eprints.utm.my/id/eprint/78794/1/MustafaAnisElPFS2016.pdf application/pdf en public http://dms.library.utm.my:8080/vital/access/manager/Repository/vital:106206 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 El-Sanfaz, Mustafa Anis The probability that an element of a non-abelian group fixes a set and its applications in graph theory
description	The commutativity degree, defined as the probability that two randomly selected elements of a group commute, plays a very important role in determining the abelianness of a group. In this research, the commutativity degree is extended by finding the probability that a group element fixes a set. This probability is computed under two group actions on the set namely, the conjugate action and the regular action. The set under study consists of all commuting elements of order two of metacyclic 2-groups and dihedral groups of even order. The probabilities found turned out to depend on the cardinality of the set. The results which were obtained from the probability are then linked to graph theory, more precisely to orbit graph and generalized conjugacy class graph. It is found that the orbit graph and the generalized conjugacy class graph consist of complete graphs, empty graphs or null graphs. Moreover, some graph properties including the chromatic number, clique number, dominating number and independent number are found. In addition, the necessary condition for the orbit graph and generalized conjugacy class graph to be a null graph is examined. Furthermore, two new graphs are introduced, namely the generalized commuting graph and the generalized non-commuting graph. The generalized commuting graph of all groups in the scope of this research turns out to be a union of complete graphs or null graphs, while the generalized non-commuting graph consists of regular graphs, empty graphs or null graphs.
format	Thesis
qualification_name	Doctor of Philosophy (PhD.)
qualification_level	Doctorate
author	El-Sanfaz, Mustafa Anis
author_facet	El-Sanfaz, Mustafa Anis
author_sort	El-Sanfaz, Mustafa Anis
title	The probability that an element of a non-abelian group fixes a set and its applications in graph theory
title_short	The probability that an element of a non-abelian group fixes a set and its applications in graph theory
title_full	The probability that an element of a non-abelian group fixes a set and its applications in graph theory
title_fullStr	The probability that an element of a non-abelian group fixes a set and its applications in graph theory
title_full_unstemmed	The probability that an element of a non-abelian group fixes a set and its applications in graph theory
title_sort	probability that an element of a non-abelian group fixes a set and its applications in graph theory
granting_institution	Universiti Teknologi Malaysia, Faculty of Science
granting_department	Faculty of Science
publishDate	2016
url	http://eprints.utm.my/id/eprint/78794/1/MustafaAnisElPFS2016.pdf
_version_	1747818072921604096

The probability that an element of a non-abelian group fixes a set and its applications in graph theory

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