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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| Format: | thesis |
| Language: | eng |
| Published: |
2017
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| Subjects: | |
| Online Access: | https://ir.upsi.edu.my/detailsg.php?det=342 |
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| Summary: | 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. |
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