UPSI Digital Repository (UDRep)
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UPSI Digital Repository (UDRep)
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| Abstract : Perpustakaan Tuanku Bainun |
| 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 with the elementary abelian 2-group C₁ x C₁ as the point group
are also considered. These groups
are polycyclic since they are extensions of polycyclic groups. Using computational methods
developed before for polycyclic groups, the nonabelian tensor squares for these Bieberbach groups
are determined. The formulas for the nonabelian tensor squares of four Bieberbach groups with
cyclic point group of order two and two Bieberbach groups with the elementary abelian 2-group C₁ x
C₂ as the 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 arbitrary 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.
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