Rohaidah Masri

The nonabelian tensor square of Bieberbach group of dimension five with dihedral point group of order eight

The nonabelian tensor product was originated in homotopy theory as well as in algebraic K-theory. The nonabelian tensor square is a special case of the nonabelian tensor product where the product is defined if the two groups act on each other in a compatible way and their action are taken to be conjugation. In this paper, the computation of nonabelian tensor square of a Bieberbach group, which is a torsion free crystallographic group, of dimension five with dihedral point group of order eight is determined. Groups, Algorithms and Programming (GAP) software has been used to assist and verify the results.

The generalization of the Schur multipliers of Bieberbach groups

The Schur multiplier is the second homology group of a group. It has been found to be isomorphic to the kernel of a homomorphism which maps the elements in the exterior square of the group to the elements in its derived subgroup. Meanwhile, a Bieberbach group is a space group which is a discrete cocompact group of isometries of oriented Euclidean space. In this research, the Schur multipliers of Bieberbach groups with cyclic point group of order two of finite dimension are computed.

On some homological functors of a Bieberbach group with symmetric point group

Bieberbach groups with symmetric point group are polycyclic. The properties of the groups can be explored by computing their homological functors. In this paper, some homological functors of a Bieberbach group with symmetric point group, such as the Schur multiplier and the G-trivial subgroup of the nonabelian tensor square, are generalized up to finite dimension and are represented in the form of direct product of cyclic groups.

The central subgroup of the nonabelian tensor square of Bieberbach group of dimension three with point group C2 × C2

Bieberbach groups are crystallographic groups. By computing the central subgroup of the nonabelian tensor square of a group, the properties of the group can be determined. In this paper, the central subgroup of the nonabelian tensor square of one Bieberbach group of dimension three with point group C2 × C2 is computed. In order to compute the ∇ (S3 (3)), the derived subgroup and the abelianization of the group are first constructed.

Homological functor of a torsion free crystallographic group of dimension five with a nonabelian point group

Torsion free crystallographic groups, called Bieberbach groups, appear as fundamental groups of compact, connected, flat Riemannian manifolds and have many interesting properties. New properties of the group can be obtained by, not limited to, exploring the groups and by computing their homological functors such as nonabelian tensor squares, the central subgroup of nonabelian tensor squares, the kernel of the mapping of nonabelian tensor squares of a group to the group and many more. In this paper, the homological functor, J(G) of a centerless torsion free crystallographic group of dimension five with a nonabelian point group which is a dihedral point group is computed using commutator calculus.

The generalization of the exterior square of a Bieberbach group

The exterior square of a group is one of the homological functors which were originated in the homotopy theory. Meanwhile, a Bieberbach group is a torsion free crystallographic group. A Bieberbach group with cyclic point group of order two, C2, of dimension n can be defined as the direct product of that group of the smallest dimension with a free abelian group. Using the group presentation and commutator generating sequence, the exterior square of a Bieberbach group with point group C2 of dimension n is computed.

On Some Homological Functors of Bieberbach Group of Dimension Four with Dihedral Point Group of Order Eight

A Bieberbach group is a torsion free crystallographic group, which is an extension of a free abelian group of finite rank by a finite point group, while homological functors of a group include nonabelian tensor square, exterior square and Schur Multiplier. In this paper, some homological functors of a Bieberbach group of dimension four with dihedral point group of order eight are computed.

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

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.

The presentation of the nonabelian tensor square of a Bieberbach group of dimension five with dihedral point group

One of the homological functors of a group, is the nonabelian tensor square. It is important in the determination of the other homological functors of a group. In order to compute the nonabelian tensor square, we need to get its independent generators and its presentation. In this paper, we present the calculation of getting the presentation of the nonabelian tensor square of the group. The presentation is computed based on its independent generators by using the polycyclic method.

On the abelianization of all Bieberbach groups of dimension four with symmetric point group of order six

A torsion free crystallographic group, which is known as a Bieberbach group, has many interesting properties. The properties of the groups can be explored by computing the homological functors of the groups. In the computation of the homological functors, the abelianization of groups plays an important role. The abelianization of a group can be constructed by computing its derived subgroup. In this paper, the construction of the abelianization of all Bieberbach groups of dimension four with symmetric point group of order six are shown. Groups, Algorithms and Programming (GAP) software is used to assist the construction.