Meta-procyclic groups. > **Theorem.** _A pro- $p$ group is meta-procyclic iff it is a inverse limit of metacyclic $p$-groups._ _Proof_. Let $G$ be a meta-procyclic pro-$p$ group with normal subgroup $N$ and let $M$ be an open normal subgroup. Then $G/M$ is procyclic since both the normal subgroup $NM/M$ and the quotient $(G/M)/(NM/M)$ are cylic $p$-groups... (continue) $\square$ * I cannot see why is $(G/M)/(NM/M)$ cyclic. If necessary, I can write the complete proof.

$G/M/(NM/M)$ is a quotient of $G/N$ (by $NM/N$) which is cyclic by assumption.