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Point groups of order at most eight and their conjugacy class graphs

Point group is a type of group in chemistry, which is a collection of symmetry elements possessed by a shape or form which all pass through one point in space. The stereographic projection is used to visualize the symmetry operations of point groups. On the other hand, group theory is the study abou...

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Main Author: Abdul Rahman, Aqilah Farhana
Format: Thesis
Language:English
Published: 2017
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QA Mathematics
Online Access:http://eprints.utm.my/id/eprint/78484/1/AqilahFarhanaAbdulMFS2017.pdf
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spelling my.utm.784842018-08-26T11:56:31Z http://eprints.utm.my/id/eprint/78484/ Point groups of order at most eight and their conjugacy class graphs Abdul Rahman, Aqilah Farhana QA Mathematics Point group is a type of group in chemistry, which is a collection of symmetry elements possessed by a shape or form which all pass through one point in space. The stereographic projection is used to visualize the symmetry operations of point groups. On the other hand, group theory is the study about an algebraic structure known as a group in mathematics. This research relates point groups of order at most eight with groups in group theory. In this research, isomorphisms, matrix representations, conjugacy classes and conjugacy class graph of point groups of order at most eight are found. The isomorphism between point groups and groups in group theory are obtained by mapping the elements of the groups and by showing that the isomorphism properties are fulfilled. Then, matrix representations of point groups are found based on the multiplication table. The conjugacy classes and conjugacy class graph of point groups of order at most eight are then obtained. From this research, it is shown that point groups C1, C3, C5 and C7 are isomorphic to the groups Z1, Z3, Z5 and Z7 respectively. Next, point groups C1h = Cs = C1v = S1, S2 = Ci, C2 = D1 are isomorphic to the group Z2, point groups C4, S4 are isomorphic to the group Z4, point groups C2h = D1d, C2v = D1h, D2 are isomorphic to the group Z2 × Z2, point groups C3v, D3 are isomorphic to the group S3, point groups C6, S6, C3h = S3 are isomorphic to the group Z6. The conjugacy classes of these groups are then applied to graph theory. It is found that the conjugacy class graph for point groups C3v and D3 are empty graphs; while conjugacy class graph for point groups C4v, D4 and D2d are complete graphs. As a conclusion, point groups in chemistry can be related with group theory in term of isomorphisms, matrix representations, conjugacy classes and conjugacy class graph. 2017-04 Thesis NonPeerReviewed application/pdf en http://eprints.utm.my/id/eprint/78484/1/AqilahFarhanaAbdulMFS2017.pdf Abdul Rahman, Aqilah Farhana (2017) Point groups of order at most eight and their conjugacy class graphs. Masters thesis, Universiti Teknologi Malaysia, Faculty of Science. http://dms.library.utm.my:8080/vital/access/manager/Repository/vital:110227
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
Abdul Rahman, Aqilah Farhana
Point groups of order at most eight and their conjugacy class graphs
description Point group is a type of group in chemistry, which is a collection of symmetry elements possessed by a shape or form which all pass through one point in space. The stereographic projection is used to visualize the symmetry operations of point groups. On the other hand, group theory is the study about an algebraic structure known as a group in mathematics. This research relates point groups of order at most eight with groups in group theory. In this research, isomorphisms, matrix representations, conjugacy classes and conjugacy class graph of point groups of order at most eight are found. The isomorphism between point groups and groups in group theory are obtained by mapping the elements of the groups and by showing that the isomorphism properties are fulfilled. Then, matrix representations of point groups are found based on the multiplication table. The conjugacy classes and conjugacy class graph of point groups of order at most eight are then obtained. From this research, it is shown that point groups C1, C3, C5 and C7 are isomorphic to the groups Z1, Z3, Z5 and Z7 respectively. Next, point groups C1h = Cs = C1v = S1, S2 = Ci, C2 = D1 are isomorphic to the group Z2, point groups C4, S4 are isomorphic to the group Z4, point groups C2h = D1d, C2v = D1h, D2 are isomorphic to the group Z2 × Z2, point groups C3v, D3 are isomorphic to the group S3, point groups C6, S6, C3h = S3 are isomorphic to the group Z6. The conjugacy classes of these groups are then applied to graph theory. It is found that the conjugacy class graph for point groups C3v and D3 are empty graphs; while conjugacy class graph for point groups C4v, D4 and D2d are complete graphs. As a conclusion, point groups in chemistry can be related with group theory in term of isomorphisms, matrix representations, conjugacy classes and conjugacy class graph.
format Thesis
author Abdul Rahman, Aqilah Farhana
author_facet Abdul Rahman, Aqilah Farhana
author_sort Abdul Rahman, Aqilah Farhana
title Point groups of order at most eight and their conjugacy class graphs
title_short Point groups of order at most eight and their conjugacy class graphs
title_full Point groups of order at most eight and their conjugacy class graphs
title_fullStr Point groups of order at most eight and their conjugacy class graphs
title_full_unstemmed Point groups of order at most eight and their conjugacy class graphs
title_sort point groups of order at most eight and their conjugacy class graphs
publishDate 2017
url http://eprints.utm.my/id/eprint/78484/1/AqilahFarhanaAbdulMFS2017.pdf
http://eprints.utm.my/id/eprint/78484/
http://dms.library.utm.my:8080/vital/access/manager/Repository/vital:110227
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score 13.211869

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