The balanced-projective dimension of abelian 𝑝-groups

Abstract

The balanced-projective dimension of every abelian p p -group is determined by means of a structural property that generalizes the third axiom of countability. As a corollary to our general structure theorem, we show for λ = ω n \lambda = {\omega _n} that every p λ {p^\lambda } -high subgroup of a p p -group G G has balanced-projective dimension exactly n n whenever G G has cardinality ℵ n {\aleph _n} but p λ G ≠ 0 {p^\lambda }G \ne 0 . Our characterization of balanced-projective dimension also leads to new classes of groups where different infinite dimensions are distinguished.