Restricted inverse zero-sum problems in groups of rank 2. (Q2890810)

Let \(G\) be a finite Abelian group of exponent \(N\), and denote by \(s(G)\) the smallest integer \(k\) such that each sequence of \(\geq k\) elements of \(G\) has a subsequence of length equal to \(N\) with vanishing sum. Replacing here equality by \(\leq N\) one obtains another constant, denoted by \(\eta(G)\). The author considers groups of rank \(2\). He presents a construction of sequences of lengths \(s(G)-1\) and \(\eta(G)-1\), respectively, which do not have such subsequences, and shows that in certain cases his construction gives all such sequences. This applies in particular to groups of the form \(C_m\oplus C_m\) in the first case, and to groups of that form with \(m=2^a3^b5^c7^d\) in the second case.