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Владикавказский математический журнал 2008, Том 10, Выпуск 1, С. 24-26
УДК 512.742
QUASI-COMPLETE Q-GROUPS ARE BOUNDED P. V. Danchev
We prove that any p-torsion quasi-complete abelian Q-group is bounded. This extends a recent statement
of ours in [6, Corollary 8] to an arbitrary large cardinality, thus also answering in the negative a conjecture
from [6]. Some other related assertions are established as well.
Mathematics Subject Classification (2000): 20K10.
Keywords: torsion-complete groups, quasi-complete groups, Q-groups, thin groups, bounded groups.
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 torsion-complete groups of cardinality N1.
Following [9], a separable group A is said to be a Q-group if for all G ^ A with |G| ^ No the inequality |(A/G)11 ^ |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 ^ 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 N1 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.
< Follows from a simple argument given in [12] and [14], respectively. >
We are now ready to attack
Theorem 2 [2]. Every quasi-complete Q-group is bounded.
< If for such a group A we have |A| ^ N1, the result was argued by us in [6] (see also the Theorem alluded to above). If now |A| > N1, we may without loss of generality assume that fin r(A) > аь 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. >
Remark. Theorem 2 resolves in a negative way the Conjecture in [6] for Q-groups.
© 2008 Danchev P. V.
Quasi-complete Q-groups are bounded
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As an immediate consequence we derive the following.
Corollary 3 [2]. A Q-group is a direct sum of quasi-complete groups if and only if it is a direct sum of cyclics.
< Write A = A¿, where, for each index i £ I, the summand A¿ is quasi-complete. Since a subgroup of a Q-group is again a Q-group [9], the foregoing theorem leads us to that every component A¿ is bounded. Therefore, A is a direct sum of cyclics. >
Conforming with [9], a group A is said to be Fuchs five if every infinite subset of A can be embedded in a direct summand of A with the same cardinality, itself.
Corollary 4. Each quasi-complete Fuchs five-group is bounded.
< It follows from [9] that separable Fuchs five-groups are themselves Q-groups. Hereafter, the preceding theorem works. >
Remark. In [10] was constructed a Q-group which is not Fuchs five. Inspired by the last result, it is of necessity not quasi-complete.
We recollect that a group A is essentially finitely indecomposable whenever A = B ® C with C a direct sum of cyclics implies that C is bounded. Likewise, a group A is known as Hi-separable provided that any countable subgroup of A is contained in a direct summand of A which is a countable direct sum of cyclics. Apparently, Hi-separable groups are separable and separable Fuchs five-groups are Hi-separable.
We are now concerned with some other similar assertions of this type presented.
Proposition 5. Each essentially finitely indecomposable Hi-separable group is bounded.
< Suppose that C is a countable subgroup of such a group A. Then C can be embedded in a direct summand of A which is a direct sum of cyclics, thus bounded. Hence C is bounded as well. Therefore all subgroups of A are bounded by a fixed positive integer and thereby A is bounded too (compare with the proof of Proposition from [6]). In fact, if A is countable we are finished. If not, i. e. A is uncountable, it possesses an uncountable number of countable subgroups whereas the number of positive integers is countable. This discrepancy allows us to conclude that A is bounded, indeed. >
According to ([8] or [13]) a group A is called essentially indecomposable if A = B ® C implies that either B or C is bounded. It is noteworthy that we have proved in [6] that -projective essentially indecomposable groups are direct sums of cyclics; we also showed in [5] that pw+i-projective essentially finitely indecomposable groups are bounded (notice that even C-decomposable pw+n-projective essentially indecomposable groups are direct sums of cyclics for each other n ^ 2 while in [1] was constructed a pw+n-projective essentially finitely indecomposable group which is not bounded — however it is clear that C-decomposable essentially finitely indecomposable groups are bounded).
We are now concentrated on a more limited class of the so-called Crawley groups [3, 4] than the class of essentially indecomposable groups. These are groups for which every direct decomposition involves a finite direct summand (for example, see also cf. [8]).
Proposition 6. Every Crawley starred group is a direct sum of cyclics.
< Owing to [11], for such a group A we may write A = C ® H where C is a direct sum of cyclics with |C| = |A|. If C is finite, then the same is true for A and we are done. Otherwise, if H is finite, it is obviously seen that A has to be a direct sum of cyclics. >
In closing, we state the following three problems of interest.
Question 1. Does it follow that each pure-complete (in particular, quasi-complete)
(a) thin group
or
26
Danchev P. V.
(b) starred group (with -CH)
or
(c) weakly H1-separable group
is a direct sum of cyclics (in particular, bounded)?
Question 2. Does it follow that each essentially finitely indecomposable (in particular, thick)
(a) thin group
or
(b) starred group (with -CH)
or
(c) weakly H1-separable group is bounded?
Question 3. What is the structure of pw+n-projective essentially finitely indecomposable Q-groups? Whether or not they are bounded?
About other questions on the same theme discussed, we refer to [6].
In accordance with [2], we shall say that the arbitrary (not necessarily p-primary) reduced abelian group K is quasi-closed if for all pure subgroups P of K the factor-group (K/P)1 is divisible. Notice that reduced algebraically compact groups are themselves quasi-closed [7]. So, we can state our final
Question 4. Determine the structure of quasi-closed Q-groups. Decide whether or not they are bounded.
References
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Received May 29, 2007. Dr. Danchev Peter V.
Plovdiv State University «Paissii Hilendarski» Plovdiv, 4003, BULGARIA E-mail: pvdanchev@yahoo.com
