Hazzirah Izzati Mat Hassim

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.

Consistent polycyclic presentation of a Bieberbach group with a nonabelian point group

Research on the nonabelian tensor square of a group is requisite on finding the other homological functors. One of the methods to explicate the nonabelian tensor square is to ensure the presentation of the group is polycyclic and to prove its consistency. In this research, the polycyclic presentation of a Bieberbach group with the quaternion point group of order eight is shown to be consistent.

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

Space groups of a crystal expound its symmetry properties. One of the symmetry properties is the central subgroup of the nonabelian tensor square of a group. It is a normal subgroup of the group which can be ascertained by finding the abelianisation of the group and the nonabelian tensor square of the abelianisation group. In this research, our focus is to explicate the central subgroup of the nonabelian tensor square of the torsion free space groups of a crystal which are called the Bieberbach groups.

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.

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.

The Schur multipliers of certain Bieberbach groups with abelian point groups

The Schur multiplier of a group G is the kernel of a homomorphism κ′ from the exterior square of the group, G ∧ G to its commutator subgroup, G′ defined by κ′(g ∧ h) = [g,h] for g,h ∈ G. In this research, the Schur multipliers are computed for certain Bieberbach groups with abelian point groups. A Bieberbach group is a torsion free crystallographic group. It is an extension of a free abelian group L of finite rank by a finite group P. Here, L is known as the lattice group while P is the point group of the Bieberbach group.