Universal valued Abelian groups

The counterparts of the Urysohn universal space in the category of metric spaces and the Gurariǐ space in the category of Banach spaces are constructed for separable valued Abelian groups of fixed (finite) exponents (and for valued groups of similar type) and their uniqueness is established. Geometry of these groups, denoted by Gr(N), is investigated and it is shown that each of Gr(N)'s is homeomorphic to the Hilbert space ℓ2. Those of Gr(N)'s which are Urysohn as metric spaces are recognized. 'Linear-like' structures on Gr(N) are studied and it is proved that every separable metrizable topological vector space may be enlarged to Gr(0) with a 'linear-like' structure which extends the linear structure of the given space.