The analysis of homological functors of some torsion free crystallographic groups with symmetric point group of order six (IR)

The analysis of homological functors of some torsion free crystallographic groups with symmetric point group of order six (IR)

This study aims to analyze the homological functors of some torsion free crystallographic groups, namely Bieberbach groups, with symmetric point group of order six. The polycyclic presentations for these groups are constructed based on their matrix representations given by Crystallographic, Algorith...

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Main Author: Tan, Yee Ting
Format: thesis
Language:eng
Published: 2017
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QA Mathematics
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institution Universiti Pendidikan Sultan Idris
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language eng
topic QA Mathematics
spellingShingle QA Mathematics
Tan, Yee Ting
The analysis of homological functors of some torsion free crystallographic groups with symmetric point group of order six (IR)
description This study aims to analyze the homological functors of some torsion free crystallographic groups, namely Bieberbach groups, with symmetric point group of order six. The polycyclic presentations for these groups are constructed based on their matrix representations given by Crystallographic, Algorithms, and Tables package, followed by checking their consistency. The homological functors which include the nonabelian tensor square, the G-trivial subgroup of the nonabelian tensor square, the central subgroup of the nonabelian tensor square, the nonabelian exterior square, and the Schur multiplier are determined by using the computational method for polycyclic groups. The structures of the nonabelian tensor squares are explored and the generalization of the homological functors of these groups are developed up to n dimension. The findings reveal that the nonabelian tensor squares and the nonabelian exterior squares of these groups are nonabelian while the rest of the homological functors are abelian. Besides, the structures of the nonabelian tensor squares of some of these groups are found split while some are found non-split. Also, the generalizations of some homological functors, which are abelian, can be represented by the products of cyclic groups while for the homological functors which are nonabelian, their generalized presentation are constructed. In conclusion, based on the formulation of the homological functors of Bieberbach groups with symmetric point group of lowest dimension, the homological functors can be generalized up to n dimension. As the implication, this study contributes new theoretical results to the field of theoretical and computational group theory and also benefit some chemists and physicists who are interested in crystallography and spectroscopy.
format thesis
qualification_name
qualification_level Doctorate
author Tan, Yee Ting
author_facet Tan, Yee Ting
author_sort Tan, Yee Ting
title The analysis of homological functors of some torsion free crystallographic groups with symmetric point group of order six (IR)
title_short The analysis of homological functors of some torsion free crystallographic groups with symmetric point group of order six (IR)
title_full The analysis of homological functors of some torsion free crystallographic groups with symmetric point group of order six (IR)
title_fullStr The analysis of homological functors of some torsion free crystallographic groups with symmetric point group of order six (IR)
title_full_unstemmed The analysis of homological functors of some torsion free crystallographic groups with symmetric point group of order six (IR)
title_sort analysis of homological functors of some torsion free crystallographic groups with symmetric point group of order six (ir)
granting_institution Universiti Pendidikan Sultan Idris
granting_department Fakulti Sains dan Matematik
publishDate 2017
url https://ir.upsi.edu.my/detailsg.php?det=342
_version_ 1747832876442845184
spelling oai:ir.upsi.edu.my:3422020-02-27 The analysis of homological functors of some torsion free crystallographic groups with symmetric point group of order six (IR) 2017 Tan, Yee Ting QA Mathematics This study aims to analyze the homological functors of some torsion free crystallographic groups, namely Bieberbach groups, with symmetric point group of order six. The polycyclic presentations for these groups are constructed based on their matrix representations given by Crystallographic, Algorithms, and Tables package, followed by checking their consistency. The homological functors which include the nonabelian tensor square, the G-trivial subgroup of the nonabelian tensor square, the central subgroup of the nonabelian tensor square, the nonabelian exterior square, and the Schur multiplier are determined by using the computational method for polycyclic groups. The structures of the nonabelian tensor squares are explored and the generalization of the homological functors of these groups are developed up to n dimension. The findings reveal that the nonabelian tensor squares and the nonabelian exterior squares of these groups are nonabelian while the rest of the homological functors are abelian. Besides, the structures of the nonabelian tensor squares of some of these groups are found split while some are found non-split. Also, the generalizations of some homological functors, which are abelian, can be represented by the products of cyclic groups while for the homological functors which are nonabelian, their generalized presentation are constructed. In conclusion, based on the formulation of the homological functors of Bieberbach groups with symmetric point group of lowest dimension, the homological functors can be generalized up to n dimension. As the implication, this study contributes new theoretical results to the field of theoretical and computational group theory and also benefit some chemists and physicists who are interested in crystallography and spectroscopy. 2017 thesis https://ir.upsi.edu.my/detailsg.php?det=342 https://ir.upsi.edu.my/detailsg.php?det=342 text eng closedAccess Doctoral Universiti Pendidikan Sultan Idris Fakulti Sains dan Matematik Adnin Afifi Nawi, Nor Muhainiah Mohd Ali, Nor Haniza Sarmin, & Samad Rashid(2016). The Schur multiplier of pairs of nonabelian groups of order p4. JurnalTeknologi, 78(3-2), 3943. doi:10.11113/jt.v78.7810Bacon, M. R. (1994). On the nonabelian tensor square of a nilpotentgroup of class two. Glasgow Mathematical Journal, 36(3), 291296.doi:10.1017/S0017089500030883Bacon, M. R., & Kappe, L. C. (1993). The nonabelian tensor square of a2-generator p-group of class 2. Archiv der Mathematik, 61(6), 508516.doi:10.1007/BF01196588Bacon, M. R., & Kappe, L. C. (2003). 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