NOTES ON COUNTABLE EXTENSIONS OF pω+n-PROJECTIVES

We prove that if G is an Abelian p-group of length not exceeding ω and H is its pω+n-projective subgroup for n ∈ N ∪ {0} such that G/H is countable, then G is also pω+n-projective. This enlarges results of ours in (Arch. Math. (Brno), 2005, 2006 and 2007) as well as a classical result due to Wallace (J. Algebra, 1971). Unless we do not specify some else, by the term “group” we mean “an Abelian p-group”, written additively as is the custom when dealing with such groups, for some arbitrary but a fixed prime p. All unexplained exclusively, but however used, notions and notations are standard and follow essentially those from [7]. For instance, a group is called separable if it does not contain elements of infinite height. As usual, for any group A, Ar denotes the reduced part of A. A recurring theme is the relationship between the properties of a given group and its countable extension (see, e.g., [1]). The study in that aspect starts incidentally by Wallace [12] in order to establish a complete set of invariants for a concrete class of mixed Abelian groups. Specifically, his remarkable achievement states as follows. Theorem (Wallace, 1971). Let G be a reduced group with a totally projective subgroup H so that G/H is countable. Then G is totally projective. Since any reduced group is summable precisely when its socle is a free valuated vector space, as application of ([8], Lemma 7) one can derive the following. Theorem (Fuchs, 1977). Let G be a reduced group with a summable isotype subgroup H so that G/H is countable. Then G is summable. Without knowing then the cited attainment of Fuchs, we have proved in [1] an analogous assertion for summable groups of countable length via the usage of a more direct group-theoretical approach. In [5] was also showed via the construction of a concrete example that when the summable subgroup H is not isotype in G, G may not be summable. Likewise, in [5] (see [1] too) it was obtained the following affirmation. 2000 Mathematics Subject Classification: Primary: 20K10; Secondary: 20K25.