Covering Finite Groups by Subset Products

Abstract

Let $G$ be a finite group and let $S$ be a nonempty subset of $G$. For any positive integer $k$, let $S^k$ be the subset product given by the set $\{s_1\cdots s_k\mid s_1,\dots ,s_k\in S\}$. If there exists a positive integer $n$ such that $S^n=G$, then $S$ is said to be exhaustive. Let $e(S)$ denote the smallest positive integer $n$, if it exists, such that $S^n=G$. We call $e(S)$ the exhaustion number of the set $S$. If $S^n\ne G$ for any positive integer $n$, then $S$ is said to be non-exhaustive. In this paper, we obtain some properties of exhaustive and non-exhaustive subsets of finite groups.