On Zero-Sum Subsequences in Finite Abelian Groups

Let G be a finite abelian group and k ∈ N with $k\not |  exp(G)$. Then E_k(G) denotes the smallest integer l ∈ N such that every sequence S ∈ F(G) with |S| ≥ l has a zero-sum subsequence T with $k \not | |T|$. In this paper we prove that if G = C_n1 ⊕···⊕ C_nr is a p-group, k ∈ N with $k \not | exp(G)$ and gcd(p, k) = 1, then

E_k(G) =  \lfloor \frac{k}{k-1}\sum_{i=1}^r (n_i-1) \rfloor+ 1.