Almost completely decomposable groups and unbounded representation type.

*(English)*Zbl 1253.20057It is well known that ‘completely decomposable’ (torsion-free) groups, i.e. direct sums of subgroups of \(\mathbb Q\), are characterized up to isomorphism by their rank 1 direct summands. In spite of this fact, the structure of ‘almost completely decomposable groups’ (i.e. finite index subgroups of finite rank completely decomposable groups) can be very complicated, and deep notions and theorems are needed in order to obtain consistent information about these groups, [cf. A. Mader, Almost completely decomposable groups. Algebra, Logic and Applications 13. Amsterdam: Gordon and Breach (2000; Zbl 0945.20031)]. For instance, a subgroup \(R\) of an almost completely decomposable group \(G\) is a ‘regulating subgroup’ if it is completely decomposable and its index is the least finite index with the property that there exists a completely decomposable subgroup of this index. The intersection \(R(G)\) of all regulating subgroups of \(G\) is called the ‘regulator’ of \(G\). If \(p\) is a prime and \(m\) is a positive integer such that \(p^mG\subseteq R(G)\) then we say that \(G\) is ‘with \(p^m\)-regulator quotient’. For every partially ordered set of types \(S\) we say that the class of \(S\)-groups (i.e. almost completely decomposable groups whose critical types are in \(S\)) with \(p^m\)-regulator quotients ‘has unbounded representation type’ if there are indecomposable groups in this class of arbitrarily large (finite) rank.

In the main result of the paper it is shown that if \(p\) is a prime and \(S\) is a finite \(p\)-locally free poset of types which have some properties (which are easy to be checked) then \(S\)-groups with \(p^m\)-regulator quotients have unbounded representation type. Although some of the proofs are very technical, they are easy to read since the paper is well written and the authors added all necessary details.

In the main result of the paper it is shown that if \(p\) is a prime and \(S\) is a finite \(p\)-locally free poset of types which have some properties (which are easy to be checked) then \(S\)-groups with \(p^m\)-regulator quotients have unbounded representation type. Although some of the proofs are very technical, they are easy to read since the paper is well written and the authors added all necessary details.

Reviewer: Simion Sorin Breaz (Cluj-Napoca)

##### MSC:

20K15 | Torsion-free groups, finite rank |

20K35 | Extensions of abelian groups |

20K25 | Direct sums, direct products, etc. for abelian groups |

20K27 | Subgroups of abelian groups |

##### Keywords:

almost completely decomposable groups; torsion-free Abelian groups of finite rank; subgroups of finite index; indecomposable groups; representations; regulating subgroups
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\textit{D. M. Arnold} et al., J. Algebra 349, No. 1, 50--62 (2012; Zbl 1253.20057)

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