On the sequence with fewer subsequence sums in finite abelian groups (Q6598013)

Let \(G\) be a finite abelian group and let \(S\) be a sequence with elements from \(G\). The length of the sequence, denoted by \(|S|\), refers to the total number of elements in \(S\), including repetitions. For each sequence \(S\), let \(\Sigma(S) \subset G\) denote the set of group elements that can be represented as the sum of one or more elements from a nonempty subsequence of \(S\). This set, \(\Sigma(S)\), contains all possible sums that can be formed from the elements of \(S\) without considering the empty subsequence. It is a known result that if \(0 \notin \Sigma(S)\), then the cardinality of \(\Sigma(S)\), denoted \(|\Sigma(S)|\), satisfies \(|\Sigma(S)| \geq |S|\). This inequality illustrates that the set of sums, when zero is excluded, must have a size at least as large as the length of the original sequence. Better lower bounds for this quantity have been deeply studied by several authors.\N\NIn this work, the authors focus on sequences \(S\) for which the union \(\Sigma(S) \cup \{0\}\) satisfies \(|\Sigma(S) \cup \{0\}| \leq |S|\). This condition imposes significant constraints on the structure of the sequence and its corresponding sumset. Their results primarily investigate the special case where \(|\Sigma(S) \cup \{0\}|\) is a prime number \(p\).\N\NMore particularly, the authors show that if this condition is met -- meaning that the size of the sumset, including zero, is a prime number -- then the sequence \(S\) must generate a cyclic subgroup of \(G\) of order \(p\). Their results indicate that the structure of the group generated by \(S\) is highly restricted under these conditions, and \(S\) necessarily forms a cyclic group with exactly \(p\) elements. This finding reveals a deep interaction between the combinatorial properties of sequences and the algebraic structure of the group to which they belong.\N\NThe authors are also able to study the problem in the case that this cardinal is equal to \(q=6,8,9\) (Theorem 1.2). On each case, the authors are able to provide an explicit characterization (up to group isomorphism) of the group generated by \(S\).