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An extension of commutativity degree of finite groups | ||
| Journal of Algebra and Related Topics | ||
| دوره 13، شماره 2، اسفند 2025، صفحه 15-26 اصل مقاله (181.34 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2024.26440.1616 | ||
| نویسندگان | ||
| M. Hashemi* 1؛ S. Gorjian2 | ||
| 1Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran | ||
| 2Department of Mathematics, University Campus 2, University of Guilan, Rasht, Iran | ||
| چکیده | ||
| Let G be a finite group. The commutativity degree of G, written d(G), is defined as the ratio |{(x, y)|x, y \in G, xy = yx}|/|G|^2. In this paper, we first extend this concept of finite groups to the commutativity degree of fuzzy subgroups. Then, by using the numerical solutions of the equation xy − zu \eqiv t(mod n), we give explicit formulas for the commutativity degree of fuzzy subgroups of 2-generated groups of nilpotency class 2. Finally we show that this method also works for a large class of finite groups, including metabelian groups. | ||
| کلیدواژهها | ||
| Nilpotent groups؛ Commutativity degree؛ Fuzzy groups | ||
| مراجع | ||
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[1] H. Doostie and M. Hashemi, Fibonacci lengths involving the Wall number k(n), J. Appl. Math. Computing, (2) 20 (2006), 171-180. [2] W. H. Gustafson, What is the probability that two group elements commute?, Amer Math. Monthly, 80 (1973), 1031-1034. [3] M. Hashemi, The commutativity degree of 2-generated groups of nilpotency class 2, Ars Combinatoria, 122 (2015), 149-159. [4] M. Hashemi, M. Pirzadeh and S. A. Gorjian, The probability that the commutator equation [x, y] = g has solution in a finite group, Journal of Algebra and Related Topics, (2) 7 (2019), 47-61. [5] M. Hashemi and M. Polkouei, Some numerical results on two classes of finite groups, Journal of Algebra and Related Topics, (1) 3 (2015), 63-72. [6] D. L. Johnson, Presentations of Groups, Cambridge University Press, 1990. [7] P. Lescot, Isoclinism classes and commutativity degrees of finite groups, J. of Algebra, 177 (1995), 847-869. [8] M. Tarnauceanu, classifying fuzzy subgroups of finite non-abelian groups, Iranian journal of fuzzy system, (4) 9 (2012), 31-41. [9] M. Tarnauceanu and L. Bentea, On the number of fuzzy subgroups of finite abelian groups, Journal Fuzzy Sets and Systems, 159 (2008), 1084-1096. | ||
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