Definable homomorphisms of abelian groups in o-minimal structures

We investigate the group H of definable homomorphisms between two definable abelian groups A and B, in an o-minimal structure N. We prove the existenc e of a "large", definable subgroup of H. If H contains an infinite definabl e set of homomorphisms then some definable subgroup of B (equivalently, a d efinable quotient of A) admits a definable multiplication, making it into a field. As we show, all of this can be carried out not only in the underlyi ng structure N but also in any structure definable in N. (C) 2000 Elsevier Science B.V. All rights reserved. MSC. 03C99; 22C05; 22B15.