Tight subgroups in torsion-free Abelian groups. (Q1424041)

The paper deals with ``tight subgroups'' of torsion-free Abelian groups, namely those subgroups that are maximal with respect to being completely decomposable. Tight subgroups were first studied by \textit{K. Benabdallah}, \textit{A. Mader} and \textit{M. A. Ould-Beddi} [J. Algebra 225, No. 1, 501-516 (2000; Zbl 0949.20039)] in almost completely decomposable groups, i.e., torsion-free groups of finite rank that contain completely decomposable subgroups of finite index. The completely decomposable subgroups of least index in an almost completely decomposable group are the ``regulating subgroups'' introduced by L. Lady and fundamental to the theory of almost completely decomposable groups. Regulating subgroups can be obtained as follows. Let \(X\) be an almost completely decomposable group and \(\tau\) a type. Then \(X(\tau)=A_\tau\oplus X^\#(\tau)\) where \(A_\tau\) is either \(0\) or \(\tau\)-homogeneous completely decomposable. It was shown by L.~Lady that \(A:=\sum_\rho A_\rho\) is actually a direct sum and the completely decomposable group \(A=\bigoplus_\rho A_\rho\) is regulating in \(X\). Regulating subgroups of almost completely decomposable groups are evidently tight but there are in general many more tight subgroups than regulating subgroups. Tight subgroups need not exist in general. In fact, a homogeneous group that is not completely decomposable contains no tight subgroups (Example~2.2). A number of criteria for a completely decomposable subgroup to be tight are established. A recurring hypothesis is that the typeset satisfy the maximum condition, and it comes into play how the subgroup is embedded in the group (regular, weakly regular). Detailed results are obtained for a generalization of almost completely decomposable groups, the oddly named ``bounded completely decomposable groups'' which are torsion-free groups \(X\) of any rank that contain a completely decomposable subgroup \(A\) such that \(X/A\) is bounded. Bounded completely decomposable groups contain regulating subgroups obtained in the way described above. A sample result is Proposition~3.2. If \(X\) is a bounded completely decomposable group whose typeset satisfies the maximum condition, then every regulating subgroup of \(X\) is strongly tight. -- Conversely, it is shown that under certain conditions tight subgroups are regulating. The final section revisits tight subgroups in almost completely decomposable groups. The main motivation is an error in the paper by Benabdallah-Mader-Ould-Beddi where it is claimed that tight subgroups of almost completely decomposable groups have finite index. An example attributed to C.~Vinsonhaler disproves this claim. The false theorem is salvaged by adding suitable hypotheses. The paper contains nice results and introduces new concepts such as ``strongly tight'' and ``sharply tight''. Concepts that describe the way a subgroup is embedded such as ``(weakly, strongly, critically) regular'', ``sharply embedded'' play an important role. A list of open problems demonstrates that the subject ``generalization of almost completely decomposable groups'' is still wide open.