Cyclic extensions of free pro-\(p\) groups (Q1305047)

Suppose that \(G\) is a pro-\(p\) group containing a normal free pro-\(p\) subgroup \(N\) such that \(G/N\simeq C_{p^n}\). Then \(G\) is the fundamental group of a profinite graph of cyclic \(p\)-subgroups of order bounded by \(p^n\). Compare this result with a characterization for free-by-cyclic groups given in well-known papers by Karrass, Pietrowski, Solitar, Cohen and Scott. A counterexample to a possible pro-\(p\) analogue of this result for noncyclic extensions is presented, too.