On the torsion groups in cotorsion classes (Q2738764)

For an arbitrary group \(G\) the classes \(G^\perp\) and \(^\perp(G^\perp)\) are defined by \(G^\perp=\{X\in\text{Mod-}\mathbb{Z}\mid\text{Ext}(G,X)=0\}\) and \(^\perp(G^\perp)=\{Y\in\text{Mod-}\mathbb{Z}\mid\text{Ext}(Y,X)=0\) for all \(X\in G^\perp\}\), respectively. Recall, that \(G^\perp\) is called the cotorsion class cogenerated by \(G\) and its subclass consisting of all torsion groups is denoted by \(\mathcal{TC}(G)\). Finally, if \(R\) is a rational group, \(\mathbb{Z}\subseteq R\subseteq\mathbb{Q}\), and \((r_p)_{p\in\Pi}\) is the characteristic of \(1\in R\), then denoting \(s_p=r_p\) whenever \(r_0=0\) or \(r_p=\infty\) and \(s_p=1\) otherwise, the characteristic \(\chi^{qr}(R)=(s_p)_{p\in\Pi}\) is called the quasi-reduced characteristic of \(R\) and the corresponding type \({\mathbf t}^{qr}(R)\) is called the quasi-reduced type of \(R\). The symbol \(R_{qr}\) denotes a rational group of the type \({\mathbf t}^{qr}(R)\).NEWLINENEWLINENEWLINENow we are going to list some of the main results. If \(R\) is a rational group then \(\mathcal{TC}(R)=\mathcal{TC}(R_{qr})\) is the class of all torsion groups of the form \(T\oplus D\), where \(D\) is divisible and the reduced part \(T=\bigoplus_{p\in\Pi}T_p\) is such that \(T_p\) is bounded whenever \(r_p=\infty\) and \(T_p=0\) for almost all primes \(p\) with \(r_p\neq 0\) (Theorem 2.4). Let \(\mathfrak C\) be a class of torsion groups. Then \({\mathfrak C}=\mathcal{TC}(C)\) for some completely decomposable group \(C\) if and only if \(\mathfrak C\) is closed under epimorphic images and contains all torsion cotorsion groups, \(\bigoplus_{n\in\omega}\mathbb{Z}(p^n)\) belongs to \(\mathfrak C\) if and only if \(\mathfrak C\) contains all \(p\)-groups for all primes \(p\), if \(P\) is an infinite set of primes, then \(\bigoplus_{p\in P}\mathbb{Z}(p)\in{\mathfrak C}\) if and only if \(\bigoplus_{p\in P}T_p\in{\mathfrak C}\) for all \(p\)-groups \(T_p\in{\mathfrak C}\) and if \(\bigoplus_{p\in P}\mathbb{Z}(p)\in{\mathfrak C}\) then there exists an infinite subset \(P'\subseteq P\) such that \(\bigoplus_{p\in X}\mathbb{Z}(p)\notin{\mathfrak C}\) for all infinite subsets \(X\subseteq P'\) (Theorem 3.6). Let \(G\) be a finite rank torsionfree group such that for any pure subgroup \(H\) of \(G\) every homomorphism from \(H\) into a torsion group \(T\) extends to a homomorphism from \(G\) to \(T\) (roughly speaking \(G\) has the Torsion Extension Property (TEP)). Then \(\mathcal{TC}(G)=\mathcal{TC}(R)\) for a rational group \(R\subseteq\mathbb{Q}\) (Theorem 4.4).NEWLINENEWLINEFor the entire collection see [Zbl 0960.00043].