@article { author = {Hashemi, M. and Pirzadeh, M. and Gorjian, S. A.}, title = {The probability that the commutator equation [x,y]=g has solution in a finite group}, journal = {Journal of Algebra and Related Topics}, volume = {7}, number = {2}, pages = {47-61}, year = {2019}, publisher = {University of Guilan}, issn = {2345-3931}, eissn = {2382-9877}, doi = {10.22124/jart.2020.15554.1187}, abstract = {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.}, keywords = {GAP,Alternating groups,Symmetric groups,Nilpotent groups}, url = {https://jart.guilan.ac.ir/article_4142.html}, eprint = {https://jart.guilan.ac.ir/article_4142_b30a2ba407287566acc0fff8ba3f52db.pdf} }