A characterization of the circle group via uniqueness of roots

Abstract

It was shown in Bíró et al. (2001) [7] that every cyclic subgroup C of the circle group T admits a characterizing sequence ( u n ) of integers in the sense that u n x → 0 for some x ∈ T iff x ∈ C . More generally, for a subgroup H of a topological (abelian) group G one can define: (a) g ( H ) to be the set of all elements x of G such that u n x → 0 in G for all sequences ( u n ) of integers such that u n h → 0 in G for all h ∈ H ; (b) H to be g -closed if H = g ( H ) . We show then that an infinite compact abelian group G has all its cyclic subgroups g -closed iff G ≅ T .