# The Quartic Commutativity Degree of Dihedral Groups

## Authors

• Muhanizah Abdul Hamid School of Mathematical Sciences, College of Computing, Informatics and Mathematics, Universiti Teknologi MARA, 40450 Shah Alam, Selangor, Malaysia
• Adnin Afifi Nawi Department of Science and Mathematics, Centre for Diploma Science, Universiti Tun Hussein Onn, Pagoh Higher Education Hub, 84600, Pagoh, Johor, Malaysia

## Keywords:

Dihedral group, Quartic commutativity degree

## Abstract

The combination of group theory and probability theory was used in studying the connection between the two. In recent years, probability theory has been widely used in solving several difficult problems in group theory. The commutativity degree of a group,  is defined as the probability that two random elements in a group commute. In addition, there exist a generalization of commutativity degree of a group which is the -th power commutativity degree of a group and it is defined as the probability of the -th power of two random elements in a group commute. Some previous studies have been found for  equal to 2 and 3 and both probabilities are called as squared commutativity degree and cubed commutativity degree respectively. In this research, the -th power commutativity degree is determined for  equal to 4, called as quartic commutativity degree and some generalization formulas have been obtained. However, this research focuses only on the dihedral groups.

## References

Miller, G. (1944). A relative number of non-invariant operators in a group. Proc. Nat. Acad. Sci. USA, 30(2), 25-28.

Lescot, P. (1995). Isoclinism classes and commutativity degree of finite groups. J. Algebra, 177, 847-86.

Rusin, D. J. (1979). What is the probability that two elements of a finite group commute? Pacific Journal of Mathematics, 82, 237-247.

Erdos, P. and Turan, P. (1968). On some problems of a statistical group theory. IV. Acta Math. Acad Sci. Hungaricae, 19, 413-435.

Mohd Ali, N. M. and Sarmin, N. H. (2010). On some problems in group theory of probabilistic nature. Menemui Matematik (Discovering Mathematics), 32(2), 35-41.

Erfanian, A., Tolue, B. and Sarmin, N. H. (2011). Some considerations on the n-th commutativity degree. Ars Combinatorial Journal. 3, 495-506.

Abd Rhani, N., Mohd Ali, N. M., Sarmin, N. H. and Erfanian, A. (2019). Multiplicative degree of some finite groups. ASM Sc. J., 12, Special Issue 5, 2019 for ICoAIMS2019, 80-85.

Muhammad, H., Susilowati, L. and Fatmawati. (2020). The proof of properties of dihedral group and its commutative elements. Journal of Physics, 1494.

Zulkifli, N., Mohd Ali, N. M., Bello, M. and Nawi, A. A. (2021). The n-th coprime probability and its graph for some dihedral groups. Journal of Physics, 1988.

Abdul Hamid, M., Mohd Ali, N. M., Sarmin, N. H., Erfanian, A. and Abd Manaf, F. N. (2016). The squared commutativity degree of dihedral groups. Jurnal Teknologi, 78(3-2),45-49.

Gallian, J. A. (1986). Contemporary abstract algebra. Health & Company Lexington, Massachusetts Toronto.

Mashkouri, M. and Taeri, B. (2011). On a graph associated to a group. Bull. Malays. Math. Sci. Soc., 34(3), 553-560.

MacHale, D. (1974). How commutative can a non-commutative group Be? The Mathematical Gazette. 58(405), 199-202.

Mohd Ali, N. M., Isah, S. I. and Bello, M. (2022). the order prime product probability and commutativity degree for some finite groups. Malaysian Journal of Mathematical Sciences, 16(4), 673-681.