Formulation of the nonabelian tensor squares of some bieberbach groups with point group C2XC2: Abelian and nonabelian cases

The main objective of this research is to compute the nonabelian tensor squares of some Bieberbach groups with elementary abelian 2-groups point groups, C2×C2 for both abelian and nonabelian cases. This is qualitative research where the computational method for polycyclic groups is used to determine the nonabelian tensor squares of four Bieberbach groups with point group C2 × C2. The findings show that the structures of the nonabelian tensor squares of some of these groups for both abelian and nonabelian cases are found split while some are found non-split. The formulas of the abelian cases of the nonabelian tensor squares of the groups lead to the construction of the formulations of the nonabelian tensor squares of arbitrary dimension. Furthermore, for the nonabelian cases, the presentations of the nonabelian tensor squares of the other two groups are given and it was shown that the nonabelian tensor squares can be written as a direct product with the nonabelian exterior square as one of the factor. As a conclusion, the formulas of the nonabelian tensor squares of four Bieberbach groups with point group C2 × C2 are developed based on its group presentations of lowest dimension and the formulas of the abelian cases can be generalized up to dimension n. As the implication, this research contributes new theoretical results in the field of theoretical and computational group theory on computing the nonabelian tensor squares of Bieberbach groups with elementary abelian 2-group point group. The results in this research will also benefit to other group theorist who are interested of the computation of the computation of the homological functors.