Two zero-sum invariants on finite abelian groups

Let G be an additive finite abelian group with exponent exp(G). Let s(G) (resp. η(G)) be the smallest integer t such that every sequence of t elements (repetition allowed) from G contains a zero-sum subsequence T of length |T|=exp(G) (resp. |T|∈[1,exp(G)]). Let H be an arbitrary finite abelian group with exp(H)=m. In this paper, we show that s(Cmn⊕H)=η(Cmn⊕H)+mn−1 holds for all n≥max{m|H|+1,4|H|+2m}.