Reflexive Group Topologies on Abelian Groups

It is proved that any infinite Abelian group of infinite exponent admits a non-discrete reflexive group topology. Introduction For a topological group G, the group G of continuous homomorphisms (characters) into the torus T = {z ∈ C : |z| = 1} endowed with the compact-open topology is called the character group of G and G is named Pontryagin reflexive or reflexive if the canonical homomorphism αG : G → G, g 7→ (χ 7→ (χ, g)) is a topological isomorphism. In the article we consider the following question. Problem 1. Is any infinite Abelian group admits a non-discrete reflexive group topology? A group G with the discrete topology is denoted by Gd. The exponent of G (=the least common multiple of the orders of the elements of G) is denoted by expG. The subgroup of G which generated by an element g is denoted by 〈g〉. Following E.G.Zelenyuk and I.V.Protasov [5], we say that a sequence u = {un} in a group G is a T -sequence if there is a Hausdorff group topology on G for which un converges to zero. The group G equipped with the finest group topology with this property is denoted by (G,u). Using the method of T -sequences, they proved that every infinite Abelian group admits a complete group topology for which characters do not separate points. Using this method, we give the positive answer to Problem 1 for groups of infinite exponent. We prove the following theorem. Theorem 1. Any infinite Abelian group G such that expG = ∞ admits a nondiscrete reflexive group topology. Let G be a Borel subgroup of a Polish group X . G is called polishable if there exists a Polish group topology τ on G such that the identity map i : (G, τ) → X, i(g) = g, is continuous. A δ-neighborhood of zero in a Polish group is denoted by Uδ. Let X be a compact metrizable group and u = {un} a sequence of elements of X. We denote by su(X) the set of all x ∈ X such that (un, x) → 1. Let G be a subgroup of X . If G = su(X) we say that u characterizes G and that G is Date: may 2009. 2000 Mathematics Subject Classification. Primary 22A10, 43A40; Secondary 54H11.