Primary abelian groups and direct sums of cyclics (Q1324182)

A subspace \(K\) of the valuated vector space \(V\) is called cofree if \(V = K \oplus F\), for some free subspace \(F\). If \(S \subseteq G[p]\), then a subgroup \(A\) is said to be supported by \(S\) if \(A[p] = S\). The group \(G\) is cofree-pure-complete if every cofree subsocle is supported by a pure subgroup. For a cardinal \(\alpha\), the homomorphism \(\phi: G \to C\) is an \(\alpha\)-homomorphism if \(C\) is a direct sum of cyclics and \(\phi(p^ n G[p])\) has rank at least \(\alpha\), for all \(n < \omega\). A homomorphism is small if its kernel contains a large subgroup and \(G\) is thick, if every homomorphism from \(G\) to a direct sum of cyclics is small. Some of the main results are as follows: 1) Suppose \(n < \omega\) and \(G\) is \(p^{\omega + n}\)-projective. Then \(G\) is cofree-pure-complete iff it is a direct sum of cycles. 2) A summand of a cofree-pure-complete group is cofree-pure-complete. 3) Suppose \(G = A \oplus B\) has a summand which is a direct sum of cyclics of final rank \(\alpha\). Then either \(A\) or \(B\) has such a summand. 4) If \(G = A \oplus B\) is \(C\)-decomposable, then so is either \(A\) or \(B\). 5) If \(G\) admits an \(\alpha\)-homomorphism, then there is a group \(H\) having a summand which is a direct sum of cyclics of final rank \(\alpha\), such that \(G\) and \(H\) embed into each other. 6) A group \(G\) is thick iff it admits no \(\omega\)-homomorphisms. 7) Suppose \(n < \omega\) and \(P\) is a \(p^ n\)-bounded subgroup of \(G\). Then \(G\) admits an \(\alpha\)-homomorphism iff \(G/P\) does.