  We call a subset $A$ of the (additive) abelian group $G$ {\it
$t$-independent} if for all non-negative integers $h$ and $k$ with $h+k \leq
t$, the sum of $h$ (not necessarily distinct) elements of $A$ does not equal
the sum of $k$ (not necessarily distinct) elements of $A$ unless $h=k$ and the
two sums contain the same terms in some order. A {\it weakly $t$-independent}
set satisfies this property for sums of distinct terms. We give some exact
values and asymptotic bounds for the size of a largest $t$-independent set and
weakly $t$-independent set in abelian groups, particularly in the cyclic group
${\mathbb Z}_n$.
