The groups considered
in this paper will be abelian primary groups. For λ a fixed but arbitrary
countable limit ordinal, C. K. Megibben studied that class Cλ consisting of all
p-groups G such that GlpαG is a direct sum of countable groups for all
α < λ.
Fundamental to the development of Cλ-theory was the introduction of the
concept of a λ-basic subgroup, which generalized the familiar concept of a basic
subgroup, and the following existence theorem: A primary group G contains a λ-basic
subgroup if and only if G is a Cλ-group. This paper extends, in a natural
fashion, the concepts of Cλ-group” and λ-basic subgroup” to an arbitrary
limit ordinal λ, and considers the analogous question of existence. This is
used to examine the structure of pλ-pure subgroups of Cλ-groups for limit
ordinals λ such that λ≠β + ω for any ordinal β. For an ordinal λ of this type, if
H is a pλ-pure subgroup of the Cλ-group G then both H and G∕H are
Cλ-groups.
