Tight subgroups in almost completely decomposable groups
A completely decomposable (cd) group is a direct sum of rank one groups, that is, groups isomorphic to additive subgroups of the rationals. An almost completely decomposable (acd) group is a torsion-free Abelian group that contains a completely decomposable subgroup of finite index. If \(A\) is a cd subgroup of the acd group \(X\) for which \(|X/A|\) is minimal, then \(A\) is called a regulating subgroup and \(\text{rgi}(X)=|X/A|\) is the regulating index. In this case the set of types of rank one summands of \(A\) is called the critical typeset of \(X\). If \(A\) is a cd subgroup that is maximal with respect to containment, then \(A\) is called a tight subgroup of \(X\). The first notable result is a simple characterization of tight subgroups. Proposition 2.7. Let \(X\) be an acd group and \(A\) a completely decomposable subgroup of finite index. Then \(A\) is tight in \(X\) if and only if every rank one summand of \(A\) is pure in \(X\). As one might expect, the index of a tight subgroup bears a relationship to the regulating index. Corollary 2.10. Let \(X\) be an acd group and \(A\) a tight subgroup. Then the prime factors of \([X:A]\) divide \(\text{rgi}(X)\) and \(\text{rgi}(X)\) divides \([X:A]\). It is shown how to construct examples of tight subgroups that are not regulating. Then the intersection of all tight subgroups is characterized in terms of the Burkhardt invariants of the acd group \(X\): \(\beta_\rho(X)=\text{rgi}(X^\#(\rho))\). Here \(X^\#(\rho)\) is the pure subgroup generated by the elements of \(X\) of type greater than \(\rho\). Corollary 3.3. Let \(X\) be an acd group. Then the intersection of all of the tight subgroups of \(X\) is \(\text{Core}(X)=\sum\{X(\rho):\rho\in T_{\text{cr}}(X)\) and \(\beta_\rho(X)=1\}\). All tight subgroups are regulating only in very special cases. Corollary 3.4. Every tight subgroup of an acd group \(X\) is a regulating subgroup if and only if \(X\) has a unique regulating subgroup. A final section discusses the quotients \(X/A\), where \(X\) is an acd group and \(A\) a tight subgroup.
scientific article; zbMATH DE number 1423624