Compact groups containing dense pseudocompact subgroups without non-trivial convergent sequences

Let G be compact abelian group such that w ( C ( G ) ) = w ( C ( G ) ) ω . We prove that if | C ( G ) | ⩾ m ( G / C ( G ) ) , then G contains a dense pseudocompact subgroup without non-trivial convergent sequences, where C ( G ) is the component of the identity of G and m ( G ) is the smallest cardinality of a dense pseudocompact subgroup of G. As a consequence we obtain the following:(1) compact connected abelian group of weight κ with κ = κ ω has a dense pseudocompact subgroup without non-trivial convergent sequences. Let G be a compact abelian group whose connected component has weight κ with κ = κ ω . The following assertions are then equivalent:(i) dense pseudocompact subgroup of G has a non-trivial convergent sequence. the following two conditions is satisfied:(a) me n < ω , nG is infinite and cf ( w ( n G ) ) = ω . G ) | < m ( log ( r 0 ( G / C ( G ) ) ) ) .