The homological functor of a Bieberbach group with a cyclic point group of order two

Abstract

Abstract. The generalized presentation of a Bieberbach group with cyclic point group of order two can be obtained from the fact that any Bieberbach group of dimension n is a direct product of the group of the smallest dimension with a free abelian group. In this paper, by using the group presentation, the homological functor of a Bieberbach group a with cyclic point group of order two of dimension n is found.

Formulation of the Nonabelian Tensor Square of a Bieberbach Group

“…Next, the other studies which are related to the determination of the formula of the nonabelian tensor squares of other Bieberbach sets with different spot groups have also been done by other researchers such as the dihedral set ( [7], [8]), the cyclic set of order three and five [9], the symmetric point set ([10, [11]) and the elementary abelian 2-set point group [12]. The abelian cases for the nonabelian tensor squares of Bieberbach sets can be found in [6] and these findings lead to the generalization of the formula of nonabelian tensor squares of the sets up to dimension n. Basedon the generalizations in [6], the homological functors of the Bieberbach set such as the schur multiplier, the nonabelian exterior squares can be computed up to dimension n in [9]. Since the nonabelian tensor square of S 2 (3) is found to be abelian in [12],this motivates us to study the generalization of the formula of the nonabelian tensor square of this set so that the homological functors of set S 2 (3) can be determined.…”

The central subgroup of the nonabelian tensor square of a torsion free space group

“…The nonabelian tensor square, denoted as , G G is generated by the symbols g h for all , g h G , subject to relations ( ) [3] did the same work but on a nonabelian point group namely the Bieberbach group of dimension five with dihedral point group of order eight. Furthermore, Mat Hassim [4] focused on the central subgroup of the Bieberbach groups with cyclic point groups of order three and five. In this research, the central subgroup of the nonabelian tensor square of the first Bieberbach group of dimension six with quaternion point group of order eight where 1, n denoted as 1 6 , Q is computed.…”