On the coperiodic hull of a separable \(p\)-group
It is well-known that every reduced abelian group \(G\) embeds into the group \(G^*=Ext(Q/Z,G)\) called the cotorsion hull of \(G\). In order to reveal the structure of \(G^*\), the author investigates the structure of \(T^*\), T being a separable \(p\)-group. Clearly, \(T^*=Ext(Z(p^{\infty}),T)\) and the main result of section 1 states that \(T^*\) is isomorphic to the group of all sequences of the form \((a_ 0,a_ 1+T,...,a_ i+T,...)\), \(a_ i\) being elements of the \(p\)-adic completion of \(T\) satisfying \(p(a_{i+1}+T)=a_ i+T\). The result enables to calculate easily heights and indicators of elements of \(T^*\). Section 3 is devoted to the case when \(T\) is a direct sum of cyclic groups. Then \(T^*\) is shown to be fully transitive and there is a one-to-one correspondence between fully characteristic subgroups of \(T^*\) and dual ideals of the semilattice of certain sequences. An example of a separable \(p\)-group \(T\) such that \(T^*\) is not fully transitive is presented.
scientific article; zbMATH DE number 4148419