Quasi-complete Q-groups Are Bounded

From the frontier of this paper, unless specified something else, let it be agreed that all groups into consideration are p-primary abelian for some arbitrary but fixed prime p written additively as is the custom when dealing with abelian group theory. The present short note is a contribution to a recent flurry of our results in [6]. Standardly, all notions and notations are essentially the same as those from [7]. For instance, A1 denotes the first Ulm subgroup of a group A. If A1 = 0, then A is termed separable. We shall also assume throughout that the Continuum Hypothesis (abbreviated as CH) holds fulfilled whenever we deal with torsioncomplete groups of cardinality א1. Following [9], a separable group A is said to be a Q-group if for all G 6 A with |G| > א0 the inequality |(A/G)1| 6 |G| holds. It is a routine technical exercise to verify that a subgroup of a Q-group is also a Q-group (see, e. g., [9]). Direct sums of cyclics are obviously Q-groups. Moreover, imitating [7], a reduced group A is called quasi-complete if for all pure G 6 A the quotient (A/G)1 is divisible. It is easily observed that these groups are also separable as well as they are closed with respect to direct summands. In [6] we obtained the following. Theorem [6]. Quasi-complete Q-groups of cardinality א1 are precisely the bounded ones. The goal here is to strengthen this claim by ignoring the cardinal restriction. First, we need the following preliminary technicality. Proposition 1. Each torsion-complete thin group is bounded. C Follows from a simple argument given in [12] and [14], respectively. B We are now ready to attack Theorem 2 [2]. Every quasi-complete Q-group is bounded. C If for such a group A we have |A| 6 א1, the result was argued by us in [6] (see also the Theorem alluded to above). If now |A| > א1, we may without loss of generality assume that fin r(A) > α1. So, it follows by virtue of [7, Theorem 74.8] that A is torsion-complete. On the other hand, A being a Q-group must be fully starred whence thin [9, 14]. Henceforth, the affirmation from previous Proposition 1 works. B Remark. Theorem 2 resolves in a negative way the Conjecture in [6] for Q-groups.