Products of subsets of group that equal the group

Abstract

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 \infty $ converges to the uniform probability on the group $G$. If $A = \left\{ {g \in G,\;P\left( g \right) \ne 0} \right\}$ be the carrier of the probability $P$, then ${A^n} = \left\{ {{a_1} \cdot ... \cdot {a_n},\;\;{a_1},...,{a_n} \in A} \right\}$ be the carrier of probability ${P^{(n)}}$. One of necessary and sufficient conditions for the mentioned convergence is: sequence ${A^n}$ at $n \to \infty $ stabilizes on $G$, i.e. ${A^k}$ = ${A^{k + 1}} = ... = G$ 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 $G$.