The central subgroup of the nonabelian tensor square of Bieberbach group with point group C2 C2

Abstract

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.