Meta-procyclic groups.

Question

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.

Answer 1

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