Balanced and cobalanced Butler groups (Q6377079)

All groups in this paper are (abelian) Butler groups of finite rank. An exact sequence (\(\dag)\) \(0\to A\to B\to C\to 0\) is balanced exact if for every completely decomposable group X the induced sequence \(Hom(X,C)\to Hom(X,B)\to 0\) is exact. It is known that this is equivalent to the exactness of \(0\to A(\tau)\to B(\tau)\to C(\tau)\to 0\) for any type \(\tau\) and it is shown to be equivalent to the purity of \(A+B(\tau)\) in B for arbitrary \(\tau\). A descending chain of classes of groups B(n) is defined inductively as follows. B(0) is the class of all Butler groups of finite rank, and \(A\in B(n)\) for \(n>0\) if and only if there is a balanced exact sequence (\(\dag)\) such that B is completely decomposable and \(C\in B(n-1)\). Finally, \(B(\infty)=\cap B(n)\). The author characterizes the classes B(n) as follows (Theorem 8). For \(n>0\), \(A\in B(n)\) if and only if for any types \(\tau_ 1,\tau_ 2,...,\tau_ n\) the subgroup \(\sum^{n}_{i=1}A(\tau_ i)\) is balanced in A or equivalently, for any types \(\tau_ 1,\tau_ 2,...,\tau_{n+1}\) the subgroup \(\sum^{n+1}_{i=1}A(\tau_ i)\) is pure in A. As a corollary it follows that \(A\in B(\infty)\) if and only if A is completely decomposable. Dually (\(\dag)\) is cobalanced exact if for every completely decomposable group X the induced sequence Hom(A,X)\(\to Hom(B,X)\to 0\) is exact. It is shown that this is equivalent to the exactness of \(0\to A[\tau]\to B[\tau]\to C[\tau]\to 0\) for any type \(\tau\) and further equivalent to the purity of \(A+B[\tau]\) in B for arbitrary \(\tau\). Dually to the definition of B(n), let CB(0) be the class of all Butler groups of finite rank, and \(C\in CB(n)\) for \(n>0\) if and only if there is a cobalanced exact sequence (\(\dag)\) such that B is completely decomposable and \(A\in CB(n-1)\). Again, \(CB(\infty)=\cap CB(n)\). The classes CB(n) are characterized in Theorem 15: For \(n>0\), the following are equivalent: (1) \(C\in CB(n)\), (2) for any types \(\tau_ 1,t_ 2,...,\tau_ n\) the subgroup \(\cap^{n}_{i=1}C(\tau_ i)=<c\in C:\) for all i \(type(c)\nleq \tau_ i>\), (3) for any types \(\tau_ 1,\tau_ 2,...,\tau_ n\) the subgroup \(\cap^{n}_{i=1}C[\tau_ i]\) is cobalanced in C, (4) for all types \(\tau\), \(C[\tau]\) is cobalanced in C and \(C[\tau]\in CB(n-1)\). As before, \(C\in CB(\infty)\) if and only if C is completely decomposable. The paper is not only worth reading for its main results but also for several useful technical lemmas. A number of misprints are easily spotted and corrected.