Abstract
Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least
$$|S| - m\ln |G|$$
elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least
$$|G{|^{1 - c{ \in ^l}}}$$
for certain c=c(m) and
$$ \in = \in (m) < 1$$
; we use the probabilistic method to give sharper values of c(m) and
$$ \in (m)$$
in the case when G is a vector space; and we give new proofs of related known results.
Counting subgroups of fixed order in finite abelian groups
