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Almost completely decomposable groups and unbounded representation type. (English) Zbl 1253.20057
It 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.

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
Full Text: DOI
[1] Arnold, D.M., Abelian groups and representations of partially ordered sets, CMS adv. books in math., (2000), Springer-Verlag New York · Zbl 0959.16011
[2] Arnold, D.M.; Dugas, M., Representation type of finite rank almost completely decomposable groups, Forum math., 10, 729-749, (1998) · Zbl 0960.20035
[3] D.M. Arnold, A. Mader, O. Mutzbauer, E. Solak, \((1, 3)\)-groups, Czechoslovak Math. J., in press. · Zbl 1326.20054
[4] Arnold, D.M.; Simson, D., Representations of finite partially ordered sets over discrete valuation rings, Comm. algebra, 35, 3128-3144, (2007) · Zbl 1142.16003
[5] Butler, M.C.R., A class of torsion-free abelian groups of finite rank, Proc. lond. math. soc. (3), 15, 680-698, (1965) · Zbl 0131.02501
[6] Faticoni, T.; Schultz, Ph., Direct decompositions of almost completely decomposable groups with primary regulating index, (), 233-242
[7] Frey, B.; Mutzbauer, O., Regulierende untergruppen und der regulator fast vollständig zerlegbarer gruppen, Rend. semin. mat. univ. Padova, 88, 1-23, (1992) · Zbl 0783.20031
[8] Fuchs, L., Infinite abelian groups, vol. II, (1973), Academic Press New York · Zbl 0257.20035
[9] Lady, E.L., Almost completely decomposable torsion-free abelian groups, Proc. amer. math. soc., 45, 41-47, (1974) · Zbl 0292.20051
[10] Mader, A., Almost completely decomposable groups, Algebra logic appl. ser., vol. 13, (2000), Gordon and Breach Amsterdam · Zbl 0945.20031
[11] Mader, A.; Mutzbauer, O., Almost completely decomposable groups with cyclic regulator quotient, (), 209-217
[12] Mader, A.; Strüngmann, L., Generalized almost completely decomposable groups, Rend. semin. mat. univ. Padova, 113, 47-69, (2005) · Zbl 1149.20045
[13] Mader, A.; Vinsonhaler, Ch., Almost completely decomposable groups with cyclic regulator quotient, J. algebra, 177, 463-492, (1995) · Zbl 0862.20041
[14] Mutzbauer, O.; Solak, E., \((1, 2)\)-groups with \(p^3\)-regulator quotient, J. algebra, 320, 3821-3831, (2008) · Zbl 1159.20028
[15] E. Solak, Almost completely decomposable groups of type \((1, 2)\), Doctoral dissertation, Universität Würzburg, 2007.
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