Existence of rigid indecomposable almost completely decomposable groups (Q2738760)

Given a finite Abelian group \(G\) and a completely decomposable group \(R\), does there exist an almost completely decomposable group \(X\) such that \(\text{R}(X)\cong R\) and \(X/\text{R}(X)\cong G\)? This is a fundamental problem that is particularly important if \(X\) is required to be indecomposable. The author proves (Theorem~3.3): Let \(R\) be a rigid completely decomposable group of finite rank and \(G\) a finite Abelian group of exponent \(m\). There exists an indecomposable almost completely decomposable group \(X\) with regulator \(\text{R}(X)\) isomorphic with \(R\) and regulator quotient \(X/\text{R}(X)\) isomorphic to \(G\) if and only if (i) and (ii) below hold.NEWLINENEWLINENEWLINE(i) \(\dim(G[p])<\dim(R/pR)\) for all prime divisors \(p\) of \(m\).NEWLINENEWLINENEWLINE(ii) There do not exist a non-trivial factorization \(m=m_1m_2\) into relatively prime factors and a proper decomposition \(R=R_1\oplus R_2\) such that \(R_1\) is \(m_1\) divisible and \(R_2\) is \(m_2\) divisible.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00043].