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 \(\{s_1 \cdots s_k \mid s_1, \ldots, s_k \in S\}\). 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 \(S^n \neq G\) 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.