Generalized almost completely decomposable groups.

*(English)*Zbl 1149.20045From 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.

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 |

##### Keywords:

torsion-free Abelian groups; direct sums of rank-one groups; finite essential extensions; generalized almost completely decomposable groups; subgroups of finite index; Butler decompositions; completely decomposable regulating subgroups; sharp subgroups
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\textit{A. Mader} and \textit{L. Strüngmann}, Rend. Semin. Mat. Univ. Padova 113, 47--69 (2005; Zbl 1149.20045)

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##### References:

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