The nonabelian tensor squares of certain bieberbach groups with cyclic point group of order two

Abstract

The torsion free crystallographic groups are called Bieberbach groups. These groups are extensions of a finite point group and a free abelian group of finite rank. The rank of the free abelian group is the dimension of Bieberbach group. In this research, Bieberbach groups with cyclic point group of order two and Bieberbach groups are elementary abelian 2-group C2xC2 as point group are also considered. These groups are polycyclic since the are extensions of polycyclic groups. Using computational methods developed before for polycyclic groups, the nonabelian tensor squares for these Bieberbach groups with cyclic point group of order two and two Bieberbach groups with the elementary abelian 2-group C2xC2 as point group are given.For the abelian nonabelian tensor square, the formula obtained can be extended to calculate the nonabelian tensor squares of Bieberbach groups of arbitary dimension. For the nonabelian cases, the nonabelian tensor squares of all Bieberbach groups with cyclic point group of order two and elementary abelian 2-group are nilpotent of class two and can be written as a direct product with the nonabelian exterior square as a factor. As a consequence, sufficient conditions for any group such that the nonabelian tensor square is abelian are obtained.