The central subgroup of the nonabelian tensor square of Bieberbach group with point group C2 C2
A Bieberbach group with point group C2 xC2 is a free torsion crystallographic group. A central subgroup of a nonabelian tensor square of a group G, denoted by ∇(G) is a normal subgroup generated by generator g⊗g for all g∈G and essentially depends on the abelianization of the group. In this paper, the formula of the central subgroup of the nonabelian tensor square of one Bieberbach group with point group C2 xC2 , of lowest dimension 3, denoted by S3 (3) is generalized up to n dimension. The consistent polycyclic presentation, the derived subgroup and the abelianization of group this group of n dimension are first determined. By using these presentations, the central subgroup of the nonabelian tensor square of this group of n dimension is constructed. The findings of this research can be further applied to compute the homological functors of this group.
Keywords: Bieberbach group; central subgroup; nonabelian tensor square.