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A generalization of the n^th- commutativity degree in finite groups | ||
| Computational Sciences and Engineering | ||
| مقاله 5، دوره 2، شماره 1، تیر 2022، صفحه 33-40 اصل مقاله (694.86 K) | ||
| نوع مقاله: Original Article | ||
| شناسه دیجیتال (DOI): 10.22124/cse.2022.21977.1028 | ||
| نویسندگان | ||
| Mina Pirzadeh* ؛ Mansour Hashemi | ||
| Department of pure Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran | ||
| چکیده | ||
| In this paper, we study the number of solutions of commutator equation [x^{n},y]=g in two classes of finite groups. For $g in G$ we consider $ rho^{n}_g(G)={(x,y)| x,yin G, [x^{n},y]=g}$ . Then the probability that the commutator equation [x^{n},y]=g has a solution in a finite group $G$, written , $P^{n}_g(G)$ is equal to $frac{|rho^{n}_{g}(G)|}{|G|^2}$ . By using the numerical solutions of the equation $xy - zu equiv t(bmod~n)$ we derive formulas for calculating the probability of $P^{n}_g(G)$, for some finite groups $G$ . | ||
| کلیدواژه‌ها | ||
| Finite groups؛ Nilpotent groups؛ Commutativity degree؛ GAP | ||
| مراجع | ||
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[1] Doostie, H., Hashemi, M. (2006. ).“Fibonacci lenghts involving the Wall number K(n) ”, J. Appl. Math. Computing, Vol. 20, pp. 171– 180. [2] The GAP Group, GAP,( 2020). “Groups, Algorithms and Programming”, Version 4.1.10, www.gap-system.org. [3] Gustafson, W. H. (1973). “What is the probability that two group elements commute?”,Amer Math. Monthly, Vol. 80, pp. 1031–1034. [4] Hashemi, M. (2015 ).“The commutative degree on 2-generated groups with nilpotency class 2”, Ars Combin, Vol. 122, pp. 149– 159. [5] Hashemi, M., Pirzadeh, M., and Gorjian, S. A., “The probability that the commutator equation [x,y]=g has solution in a finite group”, Journal of Algebra and Related Topics, Vol. 7, No. 2, pp. 47–61, 2020. [6] Pournaki, M. R., Sobhani, R (2008 ). “Probability that the commutator of two group elements is equal to a given element”, J. Pure and Applied Algebra, Vol. 212, pp. 727–734. [7] Johnson, D. L. Presentations of Groups (1997). Cambridge University Press. | ||
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