Determination of symmetry groups of 2-generator 2- groups of class two and their rationalization in molecular vibration

Abstract

Symmetry groups are powerful tools for describing the structure in physical system. The symmetry group which is composed of reflections and rotations is known as a point group. The space group of a crystal or crystallographic group is a mathematical description of the symmetry inherent in the structure. A crystallographic group G is a group extension of a group P by a free abelian group V of finite rank d. Hence there is a short exact sequence of the form 0 → V → G → P → 1. The group V is called the lattice subgroup and P is called the point group of G. The rank d of V is called the dimension the G. The Bieberbach groups are torsion free crystallographic groups. In this research, the Bieberbach groups with cyclic point group C2 up to six dimensions are considered. These groups are metabelian polycyclic groups. From the computation, there are exactly 11 non-isomorphic families Bieberbach groups when P is cyclic group of order 2 (up to dimension 6) and their group representations, which are consistent to the polycyclic representations, are obtained. Using GAP software, the matrix groups of these families are converted into finitely presented groups and polycyclic groups. These groups are investigated based on their nonabelian tensor square. By using a theory for computing the nonabelian tensor squares of polycyclic groups developed by Blyth and Morse (2008), some of properties of nonabelian tensor squares of these groups have been obtained and successfully and generalized to n dimension.