Abstract
The coprime probability of a finite group has been defined, and the coprime probability of a p-group for any prime number p has also been obtained. In this paper, we refine the concept by introducing the coprime degree of a finite group. We calculate the coprime degrees of cyclic groups, nilpotent groups, dihedral groups, and general quaternion groups.
Full text article
References
G. Miller, “Relative number of non-invariant operators in a group,” Proceedings of the National Academy of Sciences of the United States of America, vol. 30, no. 2, pp. 25–28, 1944.
P. Erd˝os and P. Tur´an, “On some problems of a statistical group theory iv,” Acta Mathematica Hungarica, vol. 19, no. 3-4, pp. 413–435, 1968.
W. Gustafson, “What is the probability that two group elements commute?,” The American Mathematical Monthly, vol. 80, no. 9, pp. 1031–1034, 1973. https://doi.org/10.2307/2318778.
D. MacHale, “How commutative can a non-commutative group be?,” The Mathematical Gazette, vol. 58, no. 405, pp. 199–202, 1974. https://doi.org/10.2307/3615961.
M. Pournaki and R. Sobhani, “Probability that the commutator of two group elements is equal to a given element,” Journal of Pure and Applied Algebra, vol. 212, no. 4, pp. 727–734, 2008. https://doi.org/10.1016/j.jpaa.2007.06.013.
R. Kanti and A. Kumar, “On a lower bound of commutativity degree,” Rendiconti del Circolo Matematico di Palermo, vol. 59, pp. 137–142, 2010. https://doi.org/10.1007/s12215-010-0010-6.
K. Moradipour, N. Sarmin, and A. Erfanian, “The precise value of commutativity degree in some finite groups,” Malaysian Journal of Fundamental and Applied Sciences, vol. 8, no. 2, pp. 67–72, 2012. https://doi.org/10.11113/mjfas.v8n2.125.
R. Nath, “Commutativity degree of a class of finite groups and consequences,” Bulletin of the Australian Mathematical Society, vol. 88, no. 3, pp. 448–452, 2013. https://doi.org/10.1017/S0004972712001086.
N. Mohd Ali, S. Isah, and M. Bello, “The order product prime probability and commutativity degree for some finite groups,” Malaysian Journal of Mathematical Sciences, vol. 16, no. 4, pp. 673–681, 2022. https://doi.org/10.47836/mjms.16.4.02.
N. Rhani, N. Ali, N. Sarmin, and A. Erfanian, “The co-prime probability of p-groups,” Journal of Mathematics and Computer Science, vol. 5, no. 1, pp. 53–56, 2019. https://ir.uitm.edu.my/id/eprint/29118/1/29118.pdf.
N. Zulkifli and N. Mohd Ali, “Co-prime probability for nonabelian metabelian groups of order less than 24 and their related graphs,” MATEMATIKA: Malaysian Journal of Industrial and Applied Mathematics, vol. 35, no. 3, pp. 357–369, 2019. https://doi.org/10.11113/matematika.v35.n3.1130.
D. Dummit and R. Foote, Abstract Algebra, 3rd ed. Wiley, 2004.
I. Isaacs, Algebra: A Graduate Course. Brooks/Cole, 1994.
D. Johnson, Topics in the Theory of Group Presentations. No. 42 in London Mathematical Society Lecture Note Series, Cambridge Univ. Pr., 1980.
Authors
Copyright (c) 2026 Journal of the Indonesian Mathematical Society
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Article Details
