Pro-\(p\) groups with waists. (Q427751)

Let \(G\) be a pro-\(p\) group. A nontrivial (open) normal \(W\) of \(G\) is defined to be waist if it is comparable with any other (open) normal subgroup \(N\), that is, either \(W\leq N\), or \(W\geq N\). This definition is modeled on that given by \textit{M.~Auslander, E.~L.~Green} and \textit{I. Reiten} for modules [Ill. J. Math. 19, 467-478 (1975; Zbl 0304.16007)].NEWLINENEWLINE The authors investigate the relation between a waist and the terms of central series. They show among others that a waist is a term of every central series with elementary Abelian sections. When the prime \(p\) is odd, they show that a waist is a term of the lower central series, unless \(G\) is procyclic, or the waist has order \(p\). With the same exceptions, if \(G\) is finite, then a waist is a term of the upper central series.