Products of subsets of group that equal the group

Let $P$ be a probability on a finite group $G$, $P^{(n)}$ $n$-fold convolution of $P$ on $G$. Under mild condition, $P^{(n)}$ at $n\to \mathrm{\infty }$ converges to the uniform probability on the group $G$. If $A=\left\{g\in G,P\left(g\right)\mathrm{\ne }0\right\}$ be the carrier of the probability $P$, then $A^n=\left\{a_1\cdot \mathrm{.}\mathrm{.}\mathrm{.}\cdot a_n,a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_n\in A\right\}$ be the carrier of probability $P^{(n)}$. One of necessary and sufficient conditions for the mentioned convergence is: sequence $A^n$ at $n\to \mathrm{\infty }$ stabilizes on $G$, i.e. $A^k$ = $A^{k+1}=\mathrm{.}\mathrm{.}\mathrm{.}=G$ for a natural number $k$. In other words, product of some multipliers equal to $A$ is $G$. The carrier $A$ is in general case any nonempty subset of group $G$. In the paper we find a condition under which product of some subsets of $G$ is $G$.