CC BY
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Владикавказский математический журнал 2009, Том 11, выпуск 3, С. 8-9
UDC 512.742
WEAKLY Ki-SEPARABLE QUASI-COMPLETE ABELIAN p-GROUPS ARE BOUNDED
P. V. Danchev
We prove that each weakly ^-separable quasi-complete abelian p-group is bounded, thus extending recent
results of ours in (Vladikavkaz Math. J., 2007 and 2008).
Mathematics Subject Classification (2000): 20K10.
Key words: weakly ^-separable groups, quasi-complete groups, torsion-complete groups, basic
subgroups, bounded groups.
Throughout the text of this brief article, suppose that all our groups are abelian p-torsion groups where p is an arbitrary prime fixed for the duration. We first of all fix two basic concepts:
Definition 1. A separable group G is called weakly Ki-separable if its each countable subgroup can be embedded in a countable pure and nice subgroup of G.
Referring to [6], G is weakly Ki-separable if and only if for every C ^ G with |C| ^ Ko the inequality |(G/C)1| ^ |C| holds. If it is fulfilled for all infinite subgroups C, then G is said to be a Q-groupj.
Definition 2. A reduced group G is called quasi-complete if for every pure subgroup P of G the quotient (G/P)1 is divisible.
It is straightforward that quasi-complete groups are themselves separable.
All other notions and notations used here are standard and follow essentially the existing classical literature [4] (see also [1] and [2]).
In [1], we showed that every weakly Ki-separable group of cardinal Ki is quasi-complete if and only if it is bounded and, in particular, the same holds for Q-groups of cardinal Ki. Specifically, the following is true:
Theorem (2007). Let G be a group of cardinality Ki. Then G is quasi-complete weakly Ki-separable if and only if G is bounded.
Next, we extended in [2] that result for Q-groups of arbitrary cardinality. In fact, the following is valid:
Theorem (2008). Let G be a group. Then G is a quasi-complete Q-group if and only if G is bounded.
It is the purpose of this short note to generalize both the results presented to weakly Ki-separable groups of any cardinality.
So, we are in a position to formulate the following assertion, provided that the Continuum Hypothesis (CH), that is, Ki = 2N° holds.
Theorem (CH). Let G be a group. Then G is quasi-complete weakly Ki-separable if and only if G is bounded.
© 2009 Danchev P. V.
Weakly Hi-separable quasi-complete abelian p-groups are bounded
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< The sufficiency is self-evident.
As for the necessity, first of all, suppose G is of cardinal Hi. Then the claim follows in virtue of the quoted above Theorem (2007).
Let us assume that |G| > Hi. Note that since any group G is isomorphic to a direct sum B © G', where B is bounded and the rank and final rank of G' agree, we will use in our situation the terms «cardinality» and «final rank» interchangeable. So, finr(G) > Hi and applying [4, Theorem 74.8] G has to be torsion-complete. In accordance with [5] (see also [4, p. 24, Theorem 68.1]) one may write that G = B for some basic subgroup B of G, where B is its torsion completion in the p-adic topology of G. Let Bi be a countable unbounded summand of B, that is, B = Bi © B2 and |Bi| ^ H0. It is not hard to verify that G = Bi © B2, where Bi and B2 are the torsion completions of Bi and B2, respectively, in the p-adic topology of G (see [4, p. 23]). It is simple to check that a subgroup of a weakly Hi-separable group is again weakly Hi-separable as well. Therefore, Bi must be weakly Hi-separable. However, utilizing [4, p. 29, Exercise 7] we derive that |B| = |Bi|Qo = HQ0 = Hi > H0 = |Bi|. But, it is well known that Bi/Bi is divisible (see, e.g., [4]) and hence |(Bi/Bi)i| = |Bi/Bi| ^ |Bi| = H0. That is why, |Bi | = |Bi | which contradicts the inequality given above. Consequently, B does not possess an unbounded summand and thus it is bounded, hence so is G because it is reduced itself, as required. >
In closing, we state the following.
Remark. Regarding [3, p. 136, Proposition 7.1] and [3, p. 137, Conjecture 7.1], by what we have shown above, it is simply impossible in any version of set theory for a weakly H1-separable group to have a subgroup which is an unbounded torsion-complete group. Indeed, as observed above, such a subgroup will always have a torsion-complete summand of cardinal 2Qo with a countable basic subgroup. So, we will always contradict the idea of weak H1-separability which is inherited by arbitrary subgroups.
So, in conclusion, Proposition 7.1 can be extended to groups with basic subgroups of arbitrary cardinality and thereby Conjecture 7.1 is settled in the negative.
Corrections: In [2] there are three misprints. First, in Theorem 2 and Corollary 3 the abbreviation «[2]» should be written and read as «(CH)», as well as on line 3 of the proof of Theorem 2 the symbol «ai» should be «Hi».
References
1. Danchev P. V. A note on weakly ^-separable p-groups // Vladikavkaz Math. J.—2007.—Vol. 9, № 1.— P. 30-37.
2. Danchev P. V. Quasi-complete Q-groups are bounded // Vladikavkaz Math. J.—2008.—Vol. 10, № 1.— P. 24-26.
3. Danchev P. V. Generalized Dieudonne and Hill criteria // Portugaliae Math.—2008.—Vol. 65, № 1.— P. 121-142.
4. Fuchs L. Infinite Abelian Groups. I, II.—Moscow: Mir, 1974, 1977.—In Russian.
5. Kulikov L. Y. On the theory of abelian groups of arbitrary cardinality // Mat. Sb.—1945.—Vol. 16, № 2.—P. 129-162.—In Russian.
6. Megibben C. K. w1-separable p-groups // Abelian Group Theory (Oberwolfach, 1985).—New York: Gordon and Breach, 1987.—P. 117-136.
Received May 26, 2008.
Danchev Peter V.
Plovdiv State University «Paissii Hilendarski», Professor
24 Tzar Assen Street, 4000 Plovdiv, Bulgaria
E-mail: [email protected]
