The object of this paper is to determine all cases in which two or more finitely generated abelian groups have the same holomorph(l). Let G and G' be finitely generated abelian groups and let H be the holomorph of G. Then it will be shown that H is the holomorph of G' if and only if G' is an invariant maximal-abelian subgroup of H isomorphic to G. All such subgroups of H are determined. There are at most four. If G does not contain any elements of order 2, or if G has at least three independent generators of infinite order, then G itself is the only such subgroup(2). 1. Definitions. Let G be a group. If aand r are two automorphisms of G, then a-r is defined to be the automorphism such that (o-r)g =o-(rg) for all gEGG. Under this composition the automorphisms of G form a group A. Consider the set H of all pairs (g, a-), gEG, -EzA. We define a composition in H by