  Suppose that $A$, $B$ and $S$ are non-empty subsets of a finite abelian group
$G$. Then the generalized restricted sumset $$ A\stackrel{S}+B:=\{a+b:\,a\in
A,\ b\in B,\ a-b\not\in S\} $$ contains at least $$ \min\{|A|+|B|-3|S|,p(G)\}
$$ elements, where $p(G)$ is the least prime factor of $|G|$. Further, we also
have $$ |A\stackrel{S}+B|\geq \min\{|A|+|B|-|S|-2,p(G)\}, $$ provided that both
$|A|$ and $|B|$ are large with respect to $|S|$.
