Isomorphism and matrix representation of point groups


  • Wan Heng Fong Universiti Teknologi Malaysia
  • Aqilahfarhana Abdul Rahman Universiti Teknologi Malaysia
  • Nor Haniza Sarmin Universiti Teknologi Malaysia



Group, isomorphism, matrix representation, point group


In chemistry, point group is a type of group used to describe the symmetry of molecules. It is a collection of symmetry elements controlled by a form or shape which all go through one point in space, which consists of all symmetry operations that are possible for every molecule. Next, a set of number or matrices which assigns to the elements of a group and represents the multiplication of the elements is said to constitute representation of a group. Here, each individual matrix is called a representative that corresponds to the symmetry operations of point groups, and the complete set of matrices is called a matrix representation of the group. This research was aimed to relate the symmetry in point groups with group theory in mathematics using the concept of isomorphism, where elements of point groups and groups were mapped such that the isomorphism properties were fulfilled. Then, matrix representations of point groups were found based on the multiplication table where symmetry operations were represented by matrices. From this research, point groups of order less than eight were shown to be isomorphic with groups in group theory. In addition, the matrix representation corresponding to the symmetry operations of these point groups wasis presented. This research would hence connect the field of mathematics and chemistry, where the relation between groups in group theory and point groups in chemistry were shown.

Author Biographies

Wan Heng Fong, Universiti Teknologi Malaysia

Department of Mathematical Sciences, Faculty of Science

Aqilahfarhana Abdul Rahman, Universiti Teknologi Malaysia

Department of Mathematical Sciences, Faculty of Science

Nor Haniza Sarmin, Universiti Teknologi Malaysia

Department of Mathematical Sciences, Faculty of Science


Cotton, F. A. 2008. Chemical Applications of Group Theory. John Wiley & Sons.

Cracknell, A. P. 2016. Applied group theory: The commonwealth and international library: Selected readings in physics: Elsevier.

Delibas, A., Aykan, V., Turkkan, D., Akkus, H. 2013. Point groups in solid state physics i: Point group oh. American Journal of Modern Physics, 2, 2, 81-87.

Ferraro, J. R., Ziomek, J. S. 1969. Introductory group theory; and its application to molecular structures: Plenum Press.

Fong, W. H. 2005. Some Characterizations of Groups of Order 8. (Master Dissertation), Universiti Teknologi Malaysia, Malaysia.

Gallian, J. 2012. Contemporary Abstract Algebra: Nelson Education.

Willock, D. 2009. Molecular Symmetry: John Wiley & Sons.

Windle, A. H. 1977. A First Course in Crystallography: G. Bell.