  A sequence of elements of a finite group G is called a zero-sum sequence if
it sums to the identity of G. The study of zero-sum sequences has a long
history with many important applications in number theory and group theory. In
1989 Kleitman and Lemke, and independently Chung, proved a strengthening of a
number theoretic conjecture of Erdos and Lemke. Kleitman and Lemke then made
more general conjectures for finite groups, strengthening the requirements of
zero-sum sequences. In this paper we prove their conjecture in the case of
abelian groups. Namely, we use graph pebbling to prove that for every sequence
(g_k)_{k=1}^{|G|} of |G| elements of a finite abelian group G there is a
nonempty subsequence (g_k)_{k in K} such that sum_{k in K}g_k=0_G and sum_{k in
K}1/|g_k|\le 1, where |g| is the order of the element g in G.
