Lifting direct sum decompositions of bounded Abelian \(p\)-groups (Q2738762)

The main result of the paper is the following theorem. Let \(G\) be a free Abelian group and \(H\) a subgroup of \(G\) such that \(G/H\) is a bounded \(p\)-group where \(p\) is a prime. Then for any direct sum decomposition \(G/H=\bigoplus_{i\in I}K_i\), there is a direct sum decomposition \(G=\bigoplus_{i\in I}L_i\) such that the quotient map takes \(L_i\) into \(K_i\). For the proof, the author first reduces (using a standard argument due to Kaplansky) to the countable case (i.e., \(G\) is a countably generated free Abelian group) and finally uses an infinite Gaussian elimination modulo \(p\) algorithm which lifts a countable direct sum of cyclic \(p\)-groups to a basis of \(G\).NEWLINENEWLINENEWLINEA Maple V implementation of the algorithm in this paper can be found on the author's web page.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00043].