# On the generalized conjugacy class graph of some dihedral groups

## Authors

• Nurhidayah Zaid Universiti Teknologi Malaysia
• Nor Haniza Sarmin Universiti Teknologi Malaysia
• Hamisan Rahmat Universiti Teknologi Malaysia

## Keywords:

Graph theory, conjugacy class, dihedral group, commutativity degree

## Abstract

A graph is a mathematical structure which consists of vertices and edges that is used to model relations between object. In this research, the generalized conjugacy class graph is constructed for some dihedral groups to show the relation between orbits and their cardinalities. In order to construct the graph, the probability that an element of the dihedral groups fixes a set must first be obtained. The set under this study is the set of all pairs of commuting elements in the form of (a,b) where a and b are elements of the dihedral groups and the lowest common multiple of the order of the elements is two. The orbits of the set are then computed using conjugation action. Based on the results obtained, the generalized conjugacy class graph is constructed and some graph properties are also found.

## Author Biographies

### Nurhidayah Zaid, Universiti Teknologi Malaysia

Department of Mathematical Sciences, Universiti Teknologi Malaysia, 81310 UTM, Johor Bahru, Johor.

### Nor Haniza Sarmin, Universiti Teknologi Malaysia

Department of Mathematical Sciences, Universiti Teknologi Malaysia, 81310 UTM, Johor Bahru, Johor.

### Hamisan Rahmat, Universiti Teknologi Malaysia

Department of Mathematical Sciences, Universiti Teknologi Malaysia, 81310 UTM, Johor Bahru, Johor.

## References

Bertram, E. A., Herzog, M., Mann, A. 1990. On a graph related to conjugacy classes of groups. Bulletin of the London Mathematical Society. 22, 569-575.

Erdos, P., Turan, P. 1968. On some problems of statistical group theory. Acta Mathematica Academiae Scientiarum Hungaricae. 19, 413-435.

Erfanian, A., Tolue, B. 2012. Conjugate graphs of finite groups. Discrete Mathematics, Algorithms and Applications. 4(2), 35-43.

Goodman, F. M. 2003. Algebra abstract and concrete stressing symmetry. Upper Saddle River: Pearson Education.

Gustafson, W. H. 1973. What is the probability that two groups element commute. The American Mathematical Monthly. 80(9), 1031-1034.

Miller, G. A. 1944. Relative number of non-invariant operators in a group. Proceeding of the National Academy of Sciences. 30(2), 25-28.

Neumann, H. 1967. Varieties of groups. Germany: Springer-Verlag, Berlin.

Omer, S. M. S., Sarmin, N. H., Erfanian, A. 2015. Generalized conjugacy class graph of some finite non-abelian groups. AIP Conference Proceedings. 050074, 1-5.

Omer, S. M. S., Sarmin, N. H., Moradipour, K., Erfanian, A. 2013. The probability that an element of a group fixes a set and the group act on set by conjugation. International Journal of Applied Mathematics and Statistics. 32(2), 111-117.

Tarnauceanu, M. 2009. Subgroup commutativity degree of finite groups. Journal of Algebra. 321, 2508-2520.

Tolue, B., Erfanian, A. 2013. Relative non-commuting graph of a finite group. Journal of Algebra and Its Applications. 12, 2, 1-11.