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A systematic approach to group properties using its geometric structure

The algebraic properties of a group can be explored through the relationship among its elements. In this paper, we define the graph that establishes a systematic relationship among the group elements. Let G be a finite group, the order product prime graph of a group G, is a graph having the elements...

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Main Authors: Bello, M., Ali, N. M. M., Zulkifli, N.
Format: Article
Language:English
Published: New York Business Global 2020
Subjects:
QA Mathematics
Online Access:http://eprints.utm.my/id/eprint/93988/1/NurfarahZulkifli2020_ASystematicApproachtoGroupProperties.pdf
http://eprints.utm.my/id/eprint/93988/
https://www.ejpam.com/index.php/ejpam/article/view/3587
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spelling my.utm.939882022-02-28T13:16:41Z http://eprints.utm.my/id/eprint/93988/ A systematic approach to group properties using its geometric structure Bello, M. Ali, N. M. M. Zulkifli, N. QA Mathematics The algebraic properties of a group can be explored through the relationship among its elements. In this paper, we define the graph that establishes a systematic relationship among the group elements. Let G be a finite group, the order product prime graph of a group G, is a graph having the elements of G as its vertices and two vertices are adjacent if and only if the product of their order is a prime power. We give the general presentation for the graph on dihedral groups and cyclic groups and classify finite dihedral groups and cyclic groups in terms of the order product prime graph as one of connected, complete, regular and planar. We also obtained some invariants of the graph such as its diameter, girth, independent number and the clique number. Furthermore, we used the vertex-cut of the graph in determining the nilpotency status of dihedral group. The graph on dihedral group is proven to be regular and complete only if the degree of the corresponding group is even prime power and connected for all prime power degree. It is also proven on cyclic group to be both regular, complete and connected if the group has prime power order. Additionally, the result turn out to show that any dihedral group whose order product prime graph's vertex-cut is greater than one is nilpotent. We also show that the order product prime graph is planar only when the degree of the group is three for dihedral group and less than five for cyclic group. Our final result shows that the order product prime graphs of any two isomorphic groups are isomophic. New York Business Global 2020 Article PeerReviewed application/pdf en http://eprints.utm.my/id/eprint/93988/1/NurfarahZulkifli2020_ASystematicApproachtoGroupProperties.pdf Bello, M. and Ali, N. M. M. and Zulkifli, N. (2020) A systematic approach to group properties using its geometric structure. European Journal of Pure and Applied Mathematics, 13 (1). 84- 95. ISSN 1307-5543 https://www.ejpam.com/index.php/ejpam/article/view/3587
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
Bello, M.
Ali, N. M. M.
Zulkifli, N.
A systematic approach to group properties using its geometric structure
description The algebraic properties of a group can be explored through the relationship among its elements. In this paper, we define the graph that establishes a systematic relationship among the group elements. Let G be a finite group, the order product prime graph of a group G, is a graph having the elements of G as its vertices and two vertices are adjacent if and only if the product of their order is a prime power. We give the general presentation for the graph on dihedral groups and cyclic groups and classify finite dihedral groups and cyclic groups in terms of the order product prime graph as one of connected, complete, regular and planar. We also obtained some invariants of the graph such as its diameter, girth, independent number and the clique number. Furthermore, we used the vertex-cut of the graph in determining the nilpotency status of dihedral group. The graph on dihedral group is proven to be regular and complete only if the degree of the corresponding group is even prime power and connected for all prime power degree. It is also proven on cyclic group to be both regular, complete and connected if the group has prime power order. Additionally, the result turn out to show that any dihedral group whose order product prime graph's vertex-cut is greater than one is nilpotent. We also show that the order product prime graph is planar only when the degree of the group is three for dihedral group and less than five for cyclic group. Our final result shows that the order product prime graphs of any two isomorphic groups are isomophic.
format Article
author Bello, M.
Ali, N. M. M.
Zulkifli, N.
author_facet Bello, M.
Ali, N. M. M.
Zulkifli, N.
author_sort Bello, M.
title A systematic approach to group properties using its geometric structure
title_short A systematic approach to group properties using its geometric structure
title_full A systematic approach to group properties using its geometric structure
title_fullStr A systematic approach to group properties using its geometric structure
title_full_unstemmed A systematic approach to group properties using its geometric structure
title_sort systematic approach to group properties using its geometric structure
publisher New York Business Global
publishDate 2020
url http://eprints.utm.my/id/eprint/93988/1/NurfarahZulkifli2020_ASystematicApproachtoGroupProperties.pdf
http://eprints.utm.my/id/eprint/93988/
https://www.ejpam.com/index.php/ejpam/article/view/3587
_version_ 1726791464204107776
score 13.211869

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