Isomorphism order for Abelian groups
Steve Bryant · Pacific Journal of Mathematics · 1958
In the theory of isometric embedding in metric spaces the following theorem is proved : Let M be a metric space every n + 3 points of which can be mapped isometrically into Euclidean %-space, then there exists an isometry from all of M into Euclidean w-space.Because of this theorem Euclidean %-space is said to have congruence order n + 3. Cl].L. M. Blumenthal has raised the question as to whether a notion analogous to that of congruence order could be developed for algebraic systems.In this paper a definition of isomorphism order is introduced for groups and a complete description of all Abelian groups having finite or kyperfinite isomorphism order is obtained.First a well known definition to avoid any possible misunderstanding of the use of the concept of rank.DEFINITION.A group G is said to have rank n if every finitely generated subgroup can be generated by n or fewer elements and n is the smallest natural number with this property.For convenience we introduce the following definition.DEFINITION.If k elements g l9 g 2 , * ,g k of a group G generate a subgroup of G which is isomorphic to a subgroup of a group H, we will say that g l9 g 2 , , g k are emheddable in H and that the subgroup generated by the #'s is embeddable in H.