The homological functors of a bieberbach group of dimension four with dihedral point group of order eight

Abstract

A Bieberbach group is a torsion free crystallographic group. It is an extension of a free abelian group of finite rank by a finite point group. The homological functors are originated in homotopy theory. In this research, some homological functors such asJGG and the exterior square of a Bieberbach group of dimension four with dihedral group of order eight are computed. This research is an extension to the research on finding the nonabelian tensor square of the same group. One of the methods to find the homological functors of the group is to use its nonabelian tensor square of the group. Therefore, the result of the nonabelian tensor square of the Bieberbach group of dimension four with dihedral point group of order eight is used to compute its homological functors. A software named Groups, Algorithms and Programming (GAP) is also used to identify some homological functors of a Bieberbach group of dimension four with dihedral group of order eight.