EXPLICIT COMPUTATION OF COMMUTATOR SUBGROUPS IN SIX-DIMENSIONAL TORSION-FREE BIEBERBACH GROUPS WITH QUATERNION POINT GROUP OF ORDER EIGHT
DOI:
https://doi.org/10.11113/jurnalteknologi.v88.25421Keywords:
Crystallography, Bieberbach group, polycyclic groups, commutator subgroup, quaternion point groupAbstract
The findings in the study of crystal structure and their properties, also known as crystallography, have many importance in practical applications. Based on a mathematical approach, several information on the crystallographic groups can be extracted by explicating their algebraic properties. One of the important parameters to compute the algebraic properties is by finding the commutator subgroup. This paper focuses on the Bieberbach groups, which are one of the torsion-free crystallographic groups. The Bieberbach groups of dimension six with the quaternion point group of order eight were found to be isomorphic to four polycyclic groups. However, previous studies have shown that the number of elements of the commutator subgroup for the first and second group, which consists of five elements, is found to be inaccurate with the aid of Groups, Algorithms and Programming (GAP) software. Furthermore, the commutator subgroup for the third and fourth group is yet to be computed. Thus, this paper includes an update on the computation of the commutator subgroup for the first and second groups, and the commutator subgroup for the third as well as the fourth group will be shown.
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