Countable Extensions of Torsion Abelian Groups

Suppose A is an abelian torsion group with a subgroup G such that A/G is countable that is, in other words, A is a torsion countable abelian extension of G. A problem of some group-theoretic interest is that of whether G ∈ K, a class of abelian groups, does imply that A ∈ K. The aim of the present paper is to settle the question for certain kinds of groups, thus extending a classical result due to Wallace (J. Algebra, 1981) proved when K coincides with the class of all totally projective p-groups. 1. Notions, notation and other conventions In all that follows, let A be an additively written abelian group, G its arbitrary fixed subgroup and A[p] the socle of A consisting of all elements {a : pa = 0}, p a prime. All other symbols of any character as well as the terminology not explicitly defined herein will follow those from the fundamental monograph books of L. Fuchs [6] and our bibliography [1–5]. It is long known that the second Prüfer theorem, archived in ([6, p. 118, Theorem 18.3]), says that any separable countable abelian p-group is a direct sum of cyclics. We also indicate the well-known generalizations to the last fact that every separable p-primary Σ-group (in particular separable summable or separable totally projective groups), respectively every separable p-primary σ-summable group (in particular separable summable or separable totally projective groups both of limit lengths confinal with ω), is a direct sum of cyclic groups. A further development on this theme is of Fuchs ([6, p. 24, Proposition 68.3]) who showed that if A is a separable p-group with a basic subgroup B for which A/B is countable, then A is a direct sum of cyclics; however he exploits the Prüfer’s affirmation. As a culmination of a series of such claims, in 1981, Wallace has argued in [12] the following excellent statement-improvement of the preceding one, concerning the extensions of totally projective subgroups by countable factor-groups, namely: 1991 Mathematics Subject Classification: 20K10, 20K20, 20K21.