Bieberbach groups with finite point groups

Abstract

A Bieberbach group is a torsion free crystallographic group. It is an extension of a lattice group, which is a maximal normal free abelian group of �nite rank, by a �nite point group. The main objective of this research is to compute the nonabelian tensor square of Bieberbach groups with a �nite nonabelian point group, in particular the dihedral group of order eight. Bieberbach groups in the Crystallographic AlgoRithms And Tables (CARAT) homepage were �rst explored and examples of the nonabelian tensor square of the groups were then computed by using the Groups, Algrorithms, Programming (GAP) software system. The exploration of the groups and the examples computed led to the exact characterization of the Bieberbach groups with trivial center. The centerless Bieberbach groups are interesting since they do not arise in the general construction of a Bieberbach group for a given point group. This construction has been shown to depend on the presentation of the point group. In addition, the experimental data of the computation of the nonabelian tensor square gives no insight into the structure of the tensor square such as its generators and relations. With the method developed for polycyclic groups, the nonabelian tensor square of one of the centerless Bieberbach groups with dihedral point group of order eight were manually computed. It has been demonstrated that the use of GAP helps to simplify the manual calculation. Furthermore, the computation of some homological functors of all 73 centerless Bieberbach groups with dihedral point group of order eight and of dimension at most six were explored. Lastly, some homological functors for Bieberbach groups with some other nonabelian point groups were also computed with the help of GAP.