Transformation of Matrix Presentation for Bieberbach Groups into Polycyclic Presentations

• Published in: Malaysian Journal of Fundamental and Applied Sciences
• Main Author: Rahman M.H.A.; Mohammad S.A.; Sarmin N.H.; Sarip N.M.M.M.; Muhktar S.N.; Zainal A.A.
• Format: Article
• Language: English
• Published: Penerbit UTM Press 2024
• Online Access: https://www.scopus.com/inward/record.uri?eid=2-s2.0-85212674904&doi=10.11113%2fmjfas.v20n6.3457&partnerID=40&md5=4e6ae61b2397127df8bf0bdeb30125f0

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. ©Copyright A. Rahman. This article is distributed under the termsofthe CreativeCommons Attribution License, which permits unrestricted use and redistribution provided thatthe original author and source are credited.