Vol. 43, No. 3, 1972

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Cλ-groups and λ-basic subgroups

Kyle David Wallace

Vol. 43 (1972), No. 3, 799–809

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 λβ + 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.

Mathematical Subject Classification 2000
Primary: 20K99
Received: 13 July 1971
Published: 1 December 1972
Kyle David Wallace