Transformation of Matrix Presentation for Bieberbach Groups into Polycyclic Presentations
DOI:
https://doi.org/10.11113/mjfas.v20n6.3457Keywords:
Crystallographic group, polycyclic presentations, quaternion point group, consistency relations.Abstract
A Bieberbach group is a torsion free crystallographic group that represents an extension of a free abelian lattice group by a finite point group. This research began by taking the group offered in the Crystallographic Algorithms and Tables (CARAT) package, which is in the matrix form. There are only four Bieberbach groups of dimension six to be isomorphic to the quaternion point group of order eight. In this study, three Bieberbach groups of dimension six with the quaternion point group of order eight that are considered as only the first group has been found its well-defined polycyclic presentation. Every group has eight generators that describe the group. However, the algorithm used in constructing the polycyclic presentation requires a new arbitrary generator to be added into the group. Then the consistency relations need to be checked and the polycyclic presentation is said to be a well-defined construction if it is consistent. Therefore, this study shows the construction of polycyclic presentation with the new arbitrary generator for all three groups. Furthermore, the polycyclic presentation for the second group has been proven to be consistent, which implies that the construction is well-defined.
References
Abdul Ladi, N. F., Masri, R., Mohd Idrus, N., Sarmin, N. H., & Tan, Y. T. (2017). The central subgroups of the nonabelian tensor squares of some Bieberbach groups with elementary abelian 2-group point group. Jurnal Teknologi (Sciences and Engineering), 79(7), 115–121.
Blyth, R. D., & Morse, R. F. (2009). Computing the nonabelian tensor squares of polycyclic groups. Journal of Algebra, 321, 2139–2148. https://doi.org/10.1016/j.jalgebra.2008.12.029
Xxxx Eick, B., & Nickel, W. (2008). Computing the Schur multiplicator and the nonabelian tensor square of a polycyclic group. Journal of Algebra, 320(2), 927–944. https://doi.org/10.1016/j.jalgebra.2008.02.041
Ellis, G., & Leonard, F. (1995). Computing Schur multipliers and tensor products of finite groups. Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical, 95A(2), 137–147.
Masri, R. (2009). The nonabelian tensor squares of certain Bieberbach groups with cyclic point group of order two. Universiti Teknologi Malaysia.
Mat Hassim, H. I., Sarmin, N. H., Mohd Ali, N. M., Masri, R., & Mohd Idrus, N. (2014). The homological functor of a Bieberbach group with a cyclic point group of order two. AIP Conference Proceedings, 1605(February), 672–677. https://doi.org/10.1063/1.4887670
Mohammad, S. A. (2018). The homological invariants of a Bieberbach group of dimension six with quaternion point group of order eight. Universiti Teknologi Malaysia.
Mohammad, S. A., Sarmin, N. H., & Hassim, H. I. M. (2021). Consistency relations of an extension polycyclic free abelian lattice group by quaternion point group. Journal of Physics: Conference Series, 1988(1). https://doi.org/10.1088/1742-6596/1988/1/012071
Mohammad, S. A., Sarmin, N. H., & Mat Hassim, H. I. (2015). Polycyclic presentations of the torsion free space group with quaternion point group of order eight. Jurnal Teknologi (Sciences and Engineering), 77(33), 151–156.
Mohammad, S. A., Sarmin, N. H., & Mat Hassim, H. I. (2017). Polycyclic transformations of crystallographic groups with quaternion point group of order eight. Malaysian Journal of Fundamental and Applied Sciences, 13(4), 788–791. https://doi.org/10.11113/mjfas.v13n4.752
Mohd Idrus, N. (2011). Bieberbach groups with finite point groups. Universiti Teknologi Malaysia.
Opgenorth, J., Plesken, W., & Schulz, T. (1998). Crystallographic algorithms and tables. Acta Crystallographica Section A: Foundations of Crystallography, 54(5), 517–531. https://doi.org/10.1107/S010876739701547X
Rocco, N. R. (1991). On a construction related to the non-abelian tensor square of a group. Boletim Da Sociedade Brasileira de Matemática, 22(1), 63–79. https://doi.org/10.1007/BF01244898
Soelberg, L. J. (2018). Finding torsion-free groups which do not have the unique product property. https://scholarsarchive.byu.edu/etd
Wan Mohd Fauzi, W. N. F., Mohd Idrus, N., Masri, R., & Sarmin, N. H. (2014). The nonabelian tensor square of Bieberbach group of dimension five with dihedral point group of order eight. AIP Conference Proceedings, 1605(February), 611–616. https://doi.org/10.1063/1.4887659
Tan, Y. T., Mohd. Idrus, N., Masri, R., Wan Mohd Fauzi, W. N. F., Sarmin, N. H., & Mat Hassim, H. I. (2015). The nonabelian tensor square of a Bieberbach group with symmetric point group of order six. Jurnal Teknologi, 78(1), 189–193. https://doi.org/10.11113/jt.v78.4385
Downloads
Published
Issue
Section
License
Copyright (c) 2024 Muhammad Hazwan A Rahman, Siti Afiqah Mohammad, Nor Haniza Sarmin, Nurain Mieza Mahirah Mohd Sarip, Siti Nuraishah Mukhtar, Ani Ayuni Zainal
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
