  We show that every finite abelian group $G$ occurs as the group of rational
points of an ordinary abelian variety over $\mathbb{F}_2$, $\mathbb{F}_3$ and
$\mathbb{F}_5$. We produce partial results for abelian varieties over a general
finite field $\mathbb{F}_q$. In particular, we show that certain abelian groups
cannot occur as groups of rational points of abelian varieties over
$\mathbb{F}_q$ when $q$ is large. Finally, we show that every finite cyclic
group arises as the group of rational points of infinitely many simple abelian
varieties over $\mathbb{F}_2$.
