The nonabelian tensor squares of one family of bieberbach groups with point group C2

Abstract

The nonabelian tensor square, GѳG, of a group G is generated by the symbols g ѳ h , where g,hεG subject to the relations gg'ѳh=( ⁸g’ѳg⁸h)(gѳh) and gѳhh’=(gѳh)(hgѳhh') for all g,g',h,h'εG, where g g'=gg'g-1 is the conjugation on the left. The nonabelian tensor square is a special case of the nonabelian tensor product which has its origins in homotopy theory. The Bieberbach groups are extensions of a point group and a free abelian group of finite rank. The rank of the free abelian group is the dimension of a Bieberbach group. In this study, we will compute the nonabelian tensor square of one family of Bieberbach groups with cyclic point group of order 2 and dimension 3 or, denoted by . This group is polycyclic since it is an extension of polycyclic groups. The nonabelian tensor squares are obtained using computational method developed by R. Blyth and R. F. Morse in 2008 for polycyclic groups.