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The probability that the commutator equation [x,y]=g has solution in a finite group | ||
| Journal of Algebra and Related Topics | ||
| دوره 7، شماره 2، اسفند 2019، صفحه 47-61 اصل مقاله (375.67 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2020.15554.1187 | ||
| نویسندگان | ||
| M. Hashemi* 1؛ M. Pirzadeh2؛ S. A. Gorjian3 | ||
| 1Faculty of mathematical sciences, University of Guilan. | ||
| 2Faculty of Mathematical Sciences, University of Guilan | ||
| 3University Compos 2, University of Guilan | ||
| چکیده | ||
| Let G be a finite group. For g\in G, an ordered pair $(x_1,y_1)\in G\times G$ is called a solution of the commutator equation $[x,y]=g$ if $[x_1,y_1]=g$. We consider \rho_g(G)=\{(x,y)| x,y\in G, [x,y]=g\}, then the probability that the commutator equation $[x,y]=g$ has solution in a finite group $G$, written P_g(G), is equal to \frac{|\rho_{g}(G)|}{|G|^2}. In this paper, we present two methods for the computing P_g(G). First by $GAP, we give certain explicit formulas for P_g(A_n) and P_g(S_n). Also we note that this method can be applied to any group of small order. Then by using the numerical solutions of the equation xy-zu \equiv t (mod~n), we derive formulas for calculating the probability of $\rho_g(G)$ where $G$ is a two generated group of nilpotency class 2. | ||
| کلیدواژه‌ها | ||
| GAP؛ Alternating groups؛ Symmetric groups؛ Nilpotent groups | ||
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