Group Theory analysis and formulation in rationalizing the crystallographic point groups

Group Theory is a powerful formal method for analyzing and rationalizing abstract and physical systems. In this research, group theory is used to rationalize the crystallographic point groups. This research focuses on Bieberbach group, that is a torsion free crystallographic group. Rationalizing this Bieberbach group means determining the group theory theories on constructing the group for any point group P, determining the properties of the group and computing the nonabelian tensor square of the group. In determining the theories on constructing the Bieberbach group, n—dimension Bieberbach group with any point group is considered. However, to determine the properties and to compute the nonabelian tensor square of the group, only the centerless Bieberbach group of dimension 4 with dihedral point group of order 8 is focused. Result shows that there exists a Bieberbach group for any point group P if P is finite. Moreover, the Bieberbach group of dimension 4 with dihedral point group of order 8 is proved to be centerless torsion free group and is the extension of four copies of cyclic group of infinite order by dihedral point group. It has a polycyclic presentation and its' presentation is proved to be consistent polycyclic where with these facts the nonabelian tensor square of this group is able to be computed by using the method introduced by Blyth and Morse.