Quasibases for nonseparable \(p\)-groups (Q2039355)

Let \(G\) be an abelian \(p\)-group with basic subgroup \(B=\bigoplus_{a\in A}\langle a\rangle\) for some set \(A\) of generators with quotient group \(G/B=\bigoplus_{D\in S}D\) for some set \(S\) of quasi-cyclic groups \( D\cong \mathbb Z(p^\infty)\). It is known [\textit{L. Fuchs}, Infinite abelian groups. Vol. I. New York-London: Academic Press (1970; Zbl 0209.05503); Infinite abelian groups. Vol. II. New York-London: Academic Press (1973; Zbl 0257.20035)] that a generating set \(X\) for \(G/B\) can be chosen such that \(G=\langle A\cup X\rangle \) with relations inherited from those of \(B\) and \(G/B\). A quasi-basis of \(G\) is a pair \((A,X)\) of such generating sets. It is also known that the subclass of inductive quasi-bases determines arrays \((A,X)\) that are complete invariants for a reduced separable group \(G\), but it is an unsolved problem in abelian group theory to classify quasi-bases in the case of non-separable \(G\). In this paper, the authors refine the concept of inductive quasi-basis to apply to certain reduced non-separable groups. In particular, they determine the array for the generalised Prüfer group of length \(2\omega+1\).