The notion of variance for random variables on a finite group $G$ as a numerical function is axiomatically introduced. The variance is applied to study questions of convergence of the product of random variables on $G$. In particular the following theorem is proved: if $x_1(\omega),\dots,x_n(\omega)$, are independent random variables on a group $G$ then for $z_n(\omega)=x_1(\omega),\dots,x_n(\omega)$ to converge almost everywhere the necessary and sufficient conditions are that distributions of $x_n(\omega)$ tend to the distribution concentrated on the unit of $G$ and the series of variances for the sequence $x_1(\omega),\dots,x_n(\omega),\dots$ converge.