On the Sequence with Fewer Subsequence Sums in Finite Abelian Groups

Let G be a finite abelian group and S a sequence with elements of G. Let |S| denote the length of S. Let \(\mathrm {\Sigma }(S)\subset G\) $\mathrm{\Sigma }(S)\subset G$ denote the set of group elements which can be expressed as a sum of a nonempty subsequence of S. It is known that if \(0\not \in \mathrm {\Sigma }(S)\) $0\notin \mathrm{\Sigma }(S)$ then \(|\mathrm {\Sigma }(S)|\ge |S|\) $|\mathrm{\Sigma }(S)|\ge |S|$ . In this paper, we study the sequence S satisfying \(|\mathrm {\Sigma }(S)\cup \{0\}|\le |S|\) $|\mathrm{\Sigma }(S)\cup \{0\}|\le |S|$ . We prove that if \(|\mathrm {\Sigma }(S)\cup \{0\}|\) $|\mathrm{\Sigma }(S)\cup \{0\}|$ is a prime number p, then \(\langle S\rangle \) $⟨S⟩$ is a cyclic group of p elements.