Every finite abelian group is the group of rational points of an ordinary abelian variety over f2, f3 and f5

(Communicated Pries) ABSTRACT. We show that every finite abelian group occurs as the group of rational points of an ordinary abelian variety over F2, F3 and F5. We produce partial results for abelian varieties over a general finite field Fq. In particular, we show that certain abelian groups cannot occur as groups of rational points of abelian varieties over Fq 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 F2.