The conjugation degree on a set of metacyclic 3-groups and 5-groups with their related graphs

The conjugation degree on a set is the probability that an element of a group fixes a set, whereby the group action considered is conjugation. The conjugation degree on a set is a variation of the commutativity degree of a group, which is the probability that two randomly chosen elements in a group...

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Main Author:	Amzee Zamri, Siti Norziahidayu
Format:	Thesis
Language:	English
Published:	2018
Subjects:	Q Science (General)
Online Access:	http://eprints.utm.my/id/eprint/81407/1/SitiNorziahidayuAmzeePFS2018.pdf
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id	my-utm-ep.81407
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spelling	my-utm-ep.814072019-08-23T04:07:10Z The conjugation degree on a set of metacyclic 3-groups and 5-groups with their related graphs 2018 Amzee Zamri, Siti Norziahidayu Q Science (General) The conjugation degree on a set is the probability that an element of a group fixes a set, whereby the group action considered is conjugation. The conjugation degree on a set is a variation of the commutativity degree of a group, which is the probability that two randomly chosen elements in a group commute. In this research, the presentation of metacyclic p-groups where p is an odd prime is used. Meanwhile, the set considered is a set of an ordered pair of commuting elements in the metacyclic p-groups, where p is equal to three and five, satisfying certain conditions. The conjugation degree on the set is obtained by dividing the number of orbits with the size of the set. Hence, the results are obtained by finding the elements of the group that follow the conditions of the ordered set, followed by the computation of the number of orbits of the set. In the second part of this research, the obtained results of the conjugation degree on a set are then associated with graph theory. The corresponding orbit graph, generalized conjugacy class graph, generalized commuting graph and generalized non-commuting graph are determined where a union of complete and null graphs, one complete and null graphs, one complete and null graphs with one empty and null graphs are found. Accordingly, several properties of these graphs are obtained, which include the degree of the vertices, the clique number, the chromatic number, the independence number, the girth, as well as the diameter of the graph. Furthermore, some new graphs are introduced, namely the orderly set graph, the order class graph, the generalized co-prime order graph, and the generalized non co-prime order graph, which resulted in the finding of one complete or empty graphs, a union of two complete or one complete graphs, a union of complete and empty graphs and a complete or empty graphs. Finally, several algebraic properties of these graphs are determined. 2018 Thesis http://eprints.utm.my/id/eprint/81407/ http://eprints.utm.my/id/eprint/81407/1/SitiNorziahidayuAmzeePFS2018.pdf application/pdf en public http://dms.library.utm.my:8080/vital/access/manager/Repository/vital:124954 phd doctoral Universiti Teknologi Malaysia Science
institution	Universiti Teknologi Malaysia
collection	UTM Institutional Repository
language	English
topic	Q Science (General)
spellingShingle	Q Science (General) Amzee Zamri, Siti Norziahidayu The conjugation degree on a set of metacyclic 3-groups and 5-groups with their related graphs
description	The conjugation degree on a set is the probability that an element of a group fixes a set, whereby the group action considered is conjugation. The conjugation degree on a set is a variation of the commutativity degree of a group, which is the probability that two randomly chosen elements in a group commute. In this research, the presentation of metacyclic p-groups where p is an odd prime is used. Meanwhile, the set considered is a set of an ordered pair of commuting elements in the metacyclic p-groups, where p is equal to three and five, satisfying certain conditions. The conjugation degree on the set is obtained by dividing the number of orbits with the size of the set. Hence, the results are obtained by finding the elements of the group that follow the conditions of the ordered set, followed by the computation of the number of orbits of the set. In the second part of this research, the obtained results of the conjugation degree on a set are then associated with graph theory. The corresponding orbit graph, generalized conjugacy class graph, generalized commuting graph and generalized non-commuting graph are determined where a union of complete and null graphs, one complete and null graphs, one complete and null graphs with one empty and null graphs are found. Accordingly, several properties of these graphs are obtained, which include the degree of the vertices, the clique number, the chromatic number, the independence number, the girth, as well as the diameter of the graph. Furthermore, some new graphs are introduced, namely the orderly set graph, the order class graph, the generalized co-prime order graph, and the generalized non co-prime order graph, which resulted in the finding of one complete or empty graphs, a union of two complete or one complete graphs, a union of complete and empty graphs and a complete or empty graphs. Finally, several algebraic properties of these graphs are determined.
format	Thesis
qualification_name	Doctor of Philosophy (PhD.)
qualification_level	Doctorate
author	Amzee Zamri, Siti Norziahidayu
author_facet	Amzee Zamri, Siti Norziahidayu
author_sort	Amzee Zamri, Siti Norziahidayu
title	The conjugation degree on a set of metacyclic 3-groups and 5-groups with their related graphs
title_short	The conjugation degree on a set of metacyclic 3-groups and 5-groups with their related graphs
title_full	The conjugation degree on a set of metacyclic 3-groups and 5-groups with their related graphs
title_fullStr	The conjugation degree on a set of metacyclic 3-groups and 5-groups with their related graphs
title_full_unstemmed	The conjugation degree on a set of metacyclic 3-groups and 5-groups with their related graphs
title_sort	conjugation degree on a set of metacyclic 3-groups and 5-groups with their related graphs
granting_institution	Universiti Teknologi Malaysia
granting_department	Science
publishDate	2018
url	http://eprints.utm.my/id/eprint/81407/1/SitiNorziahidayuAmzeePFS2018.pdf
_version_	1747818324364886016

The conjugation degree on a set of metacyclic 3-groups and 5-groups with their related graphs

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