On parasoluble topological groups (Q1901947)

Given closed, normal subgroups \(X\) and \(Y\) of a topological group \(G\) with \(X \subseteq Y\), the quotient \(Y/X\) is said to be \(G\)-paracentral if (a) it is Abelian and (b) every element of \(G\) induces an automorphism of \(Y/X\) which fixes every closed subgroup of \(Y/X\). The group \(G\) is said to be parasoluble if it admits a (finite) paracentral series, i.e., a series \[ e = G_1 < G_2 < \cdots < G_n = G \tag{*} \] with each \(G_{k + 1}/G_k\) paracentral; when this occurs, the least possible value of \(n\), denoted \(s(G)\), is the paraheight of \(G\). The author's aim is to replace, for a given parasoluble group \(G\), the paracentral series of length \(s(G)\) with other, perhaps longer, series whose quotients have manageable, desirable properties. Detailed statements are too technical for inclusion here, but the following results will give the flavor. 5'. One may arrange (*) with \(n \leq 2s\) and with each \(G_{k + 1}/G_k\) either connected or zero-dimensional. 5''. One may arrange (*) with \(n \leq 3s\) and with each \(G_{k + 1}/G_k\) either connected, or zero-dimensional and inductively compact, or discrete and torsion-free. Theorem 1(a). One may arrange (*) with \(n \leq 3s+1\) such that (a) each connected quotient precedes each zero-dimensional quotient and (b) every torsion quotient precedes each discrete torsion-free quotient, except that perhaps the final quotient satisfies \(|G_n/G_{n - 1} |< \omega\).