×

zbMATH — the first resource for mathematics

Generalized almost completely decomposable groups. (English) Zbl 1149.20045
From the introduction: “Rank-one groups” are torsion-free Abelian groups isomorphic to some additive subgroup of the group of rational numbers. These groups have been classified by the so-called “types” that have a concrete description. However, here a type will simply be an isomorphism class of rank-one groups. “Completely decomposable groups” are direct sums of rank-one groups and have been classified up to isomorphism. “Almost completely decomposable (acd)-groups” are finite essential extensions of completely decomposable groups of finite rank. These groups have been studied extensively during the last fifteen years and although much is known about them, it is a class of groups whose complete understanding is beyond reach. Attemps have been made to extend the concepts, ideas and results of acd-groups to infinite rank. As an example, [in D. M. Arnold, Lect. Notes Math. 874, 1-31 (1981; Zbl 0466.20030) and A. Mader, L. Strüngmann, J. Algebra 229, No. 1, 205-233 (2000; Zbl 0957.20034)] “bcd-groups” were studied that are, by definition, essential extensions of completely decomposable groups of arbitrary rank by bounded groups. The question remains what “generalized almost completely decomposable groups” should be. Now almost completely decomposable groups (of finite rank) are not only finite extensions of completely decomposable groups but are also contained in completely decomposable groups as subgroups of finite index. Hence another way of generalizing almost completely decomposable groups is to consider special subgroups of completely decomposable groups. Core features of almost completely decomposable groups are the existence of Butler decompositions and of completely decomposable regulating subgroups. Certainly these features should be present in “generalized almost completely decomposable groups”.
A subgroup \(H\) is “sharp” in \(G\) if \(H^\sharp(\tau)=H\cap G^\sharp(\tau)\) for every type \(\tau\). A main result (Theorem 4.5) characterizes almost completely decomposable groups as exactly the sharp subgroups of completely decomposable groups of any rank. This characterization makes sense for any rank and justifies our definition of “generalized almost completely decomposable groups” as the sharp subgroups of arbitrary completely decomposable groups. We show that sharp subgroups of completely decomposable groups have Butler decompositions and completely decomposable regulating subgroups (Corollary 4.1, Corollary 4.8) hence qualify to be called generalized almost completely decomposable groups. The earlier class of bcd-groups is a subclass of the new class.

MSC:
20K20 Torsion-free groups, infinite rank
20K25 Direct sums, direct products, etc. for abelian groups
20K27 Subgroups of abelian groups
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] D. M. ARNOLD, Pure subgroups of finite rank completely decomposable groups. In Proceedings of the Oberwolfach Abelian Group Theory Conference, volume 874 of Lecture Notes in Mathematics, SpringerVerlag (1981), pp. 1-31. Zbl0466.20030 MR645913 · Zbl 0466.20030
[2] D. M. ARNOLD, Finite Rank Torsion Free Abelian Groups and Rings, volume 931 of Lecture Notes in Mathematics, Springer Verlag, 1982. Zbl0493.20034 MR665251 · Zbl 0493.20034 · doi:10.1007/BFb0094245
[3] D. M. ARNOLD, Abelian Groups and Representations of Finite Partially Ordered Sets, volume 2 of CMS Books in Mathematics, Springer Verlag, 2000. Zbl0959.16011 MR1764257 · Zbl 0959.16011
[4] D. M. ARNOLD - C. VINSONHALER, Pure subgroups of finite rank completely decomposable groups II. In Proceedings of the Honolulu Abelian Groups Conference, volume 1006 of Lecture Notes in Mathematics, Springer-Verlag (1984), pp. 97-143. Zbl0522.20037 MR722614 · Zbl 0522.20037
[5] R. BAER, Abelian groups without elements of finite order, Duke Math. J., 3 (1937), pp. 68-122. MR1545974 JFM63.0074.02 · Zbl 0016.20303 · doi:10.1215/S0012-7094-37-00308-9
[6] A. ELTER, Torsionsfreie Gruppen mit linear geordneter Typenmenge, PhD thesis, Universität Essen, 1996. Zbl0893.20040 · Zbl 0893.20040
[7] J. ERDÖS, On direct decompositions of torsion free abelian groups, Publ. Math. Debrecen, 3 (1954), pp. 281-88,. Zbl0057.25604 MR72132 · Zbl 0057.25604
[8] L. FUCHS, Infinite Abelian Groups, Vol. I, II, Academic Press, 1970 and 1973. Zbl0209.05503 MR255673 · Zbl 0209.05503
[9] A. MADER, Almost Completely Decomposable Groups, volume 13 of Algebra, Logic and Applications, Gordon and Breach Science Publishers, 2000. Zbl0945.20031 MR1751515 · Zbl 0945.20031
[10] E. MÜLLER - O. MUTZBAUER, Regularität in torsionsfreien abelschen Gruppen, Czechoslovak Math. J., 42 (1992), pp. 279-288. Zbl0786.20036 MR1179499 · Zbl 0786.20036 · eudml:31275
[11] A. MADER - L. STRÜNGMANN, Bounded essential extensions of completely decomposable groups, J. Algebra, 229 (2000), pp. 205-233. Zbl0957.20034 MR1765779 · Zbl 0957.20034 · doi:10.1006/jabr.2000.8306
[12] L. G. NONGXA, *-pure subgroups of completely decomposable abelian groups, Proc. Amer. Math. Soc., 100 (1987), pp. 613-618. Zbl0624.20037 MR894425 · Zbl 0624.20037 · doi:10.2307/2046693
[13] L. PROCHÁZKA, A generalization of a theorem of R. Baer, Comment. Math. Univ. Carolinae, 4 (1963), pp. 105-108. Zbl0135.06101 MR178067 · Zbl 0135.06101 · eudml:16062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.