The probability that the commutator equation [x,y]=g has solution in a finite group

Document Type: Research Paper

Authors

1 Faculty of mathematical sciences, University of Guilan.

2 Faculty of Mathematical Sciences, University of Guilan

3 University Compos 2, University of Guilan

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