Balanced Butler groups (Q1911627)

A group \(H\) is a Butler group if there is an exact sequence \(0\to H\to G\to K\to 0\) with \(G\) completely decomposable and \(K\) torsion-free. Such a sequence is called a \({\mathcal K} (0)\)-sequence and \({\mathcal K} (0)\) is, by definition, the class of all Butler groups. \textit{A. A. Kravchenko} [Mat. Zametki 45, No. 5, 32-37 (1989; Zbl 0695.20032)] introduced a sequence of classes \[ {\mathcal K} (0)\supset {\mathcal K} (1)\supset\cdots\supset {\mathcal K} (n-1)\supset {\mathcal K} (n)\supset\cdots \] where, by definition, \(H\in {\mathcal K}(n)\) if and only if there is a balanced exact sequence \(0\to H\to G\to K\to 0\) with \(G\) completely decomposable and \(K\in {\mathcal K} (n-1)\). An exact sequence \(0\to H\to G\to K\to 0\) is balanced exact if for all types \(\tau\), the sequence \(0\to H(\tau)\to G(\tau)\to K(\tau)\to 0\) is exact. Results of Kravchenko are proved anew and differently in this paper, in particular, it is shown by example that the chain is properly decreasing and that intersection of all classes is the class of completely decomposable groups. It is a general phenomenon that theorems valid for groups in \({\mathcal K} (0)\) have stronger variants valid for groups in \({\mathcal K} (n)\). The paper contains a wealth of significant and beautiful results. Samples are the following. Theorem 1.8. (Kravchenko) Let \(H\) be an abelian group and \(n\) a non-negative integer. Then the following are equivalent. 1. \(H\in {\mathcal K}(n)\). 2. For all types \(\tau_1,\dots,\tau_n\), the subgroup \(H(\tau_1) +\cdots+H(\tau_n)\) is balanced in \(H\). 3. For all types \(\tau_1,\dots,\tau_{n+1}\), the subgroup \(H(\tau_1) +\cdots+H(\tau_{n+1})\) is pure in \(H\). Corollary 1.11 The classes \({\mathcal K} (n)\) are closed under near-isomorphism. Theorem 2.1 \(H\in {\mathcal K}(n)\) is completely decomposable if \(|T_{cr(H)}|\leq n+1\) or if \(T_{cr}(H) =\{\tau_1,\dots,\tau_{n+2}\}\) and either \(|\{\tau_i\wedge\tau_j : 1\leq i < j\leq n+2\}|\geq 2\), or \(\text{sup}\{\tau_1,\dots,\tau_{n+2}\} =\text{type }\mathbb{Q}\). Here \(T_{cr} (H)\) denotes the critical typeset of \(H\) consisting of all types \(\tau\) with \(H(\tau)\neq H^\# (\tau)\). Theorem 2.1 implies that the intersection of the classes \({\mathcal K} (n)\) is the class of completely decomposable groups. Proposition 2.3 If \(0\to H\to G\to K\to 0\) is a \({\mathcal K}(n)\)-sequence and \(|T_{cr} (G)|< {n+3\choose 2}\), then \(H\) is completely decomposable. The last section 5. contains interesting applications to almost completely decomposable groups.