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 ( C m n ⊕ H ) = η ( C m n ⊕ H ) + m n − 1 holds for all n ≥ max { m | H | + 1 , 4 | H | + 2 m } .