Characterization of almost maximally almost-periodic groups

Let G be an abelian group. We prove that a group G admits a Hausdorff group topology τ such that the von Neumann radical n(G, τ) of (G, τ) is non-trivial and finite iff G has a non-trivial finite subgroup. If G is a topological group, then n(n(G)) 6= n(G) if and only if n(G) is not dually embedded. In particular, n(n(Z, τ)) = n(Z, τ) for any Hausdorff group topology τ on Z. We shall write our abelian groups additively. For a topological group X, X̂ denotes the group of all continuous characters on X. We denote its dual group by X, i.e. the group X̂ endowed with the compact-open topology. Denote by n(X) = ∩χ∈ b Xkerχ the von Neumann radical of X. If H is a subgroup, we denote by H its annihilator. If A be a subset of a group X, 〈A〉 denotes the subgroup generated by A. Let X be a topological group and u = {un} a sequence of elements of X̂. 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 characterized (by u). Following E.G.Zelenyuk and I.V.Protasov [10], [11], 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). Group topologies on Z(p) with n(Z(p)) = Z(p) were considered in corollary 4.9 [3]. Although no explicit construction of such topology was given there, it was conjectured by D. Dikranjan that such topology can be found by means of an appropriate T -sequence on Z(p). This conjecture was successfully proved by G. Lukács [7] for every prime p 6= 2. He called a Hausdorff topological group G almost maximally almost-periodic if n(G) is non-trivial and finite and raised the problem of their description. He proved that infinite direct sums and the Prüfer group Z(p), for every prime p 6= 2, are almost maximally almost-periodic. A.P.Nguyen [8] generalized these results and proved that any Prüfer groups Z(p) and a wide class of torsion groups admit a (Hausdorff) almost maximally almost-periodic group topology. Using theorem 4 [5], we give a general characterization of almost maximally almost-periodic groups. Theorem 1. Let G be an infinite group. Then the following statements are equivalent. 1. G admits a T -sequence u such that (G,u) is almost maximally almost-periodic. ∗The author was partially supported by Israel Ministry of Immigrant Absorption