Infinite pro-$p$ group of finite solvable length and finite coclass

Question

I was reading about infinite pro-$p$ groups of finite coclass from the book "The Structure of Groups of Prime Power Order" by Leedham-Green and McKay. I asked this question in math.stackExchange before posting it here; there were no answers there, I thought I might share it in mathoverflow. My advance apologies if anything is inappropriate.

I was thinking about solvability and I think it can be shown that if $G$ is a solvable group of solvable length $l$ then every subgroup and quotient of $G$ has solvable length at most $l$ (please correct me if I am wrong). My question is related to the "opposite" of this property.

My question is

Let $S$ be an infinite pro-$p$ group of finite coclass. Suppose there exists a non-negative integer $t$ such that the solvable length of each lower central series quotient $S/\gamma_i(S)$ is less than or equal to $l$ for all $i\ge t$. Then is it true that the $S$ is solvable with solvable length less than or equal to $l$?

To recall, the coclass of a finite $p$-group $G$ of order $p^n$ is defined as $n-c$ where $c$ is the nilpotency class of $G$. In case of infinite pro-$p$ groups, an infinite pro-$p$ group $S$ is said to be of finite coclass $r$ if its lower central series quotients $S/\gamma_i(S)$ are finite $p$-groups and $S/\gamma_i(S)$ has coclass $r$ for all $i\ge t$ for some $t\ge 0$.

Thanks in advance.

Answer 1

The answer is yes, $S$ is an inverse limit of its lower central quotients. As these have bounded derived length, the same goes for the Cartesian product of these groups.

By the way, all pro-$p$ groups of finite coclass are solvable, that's Theorem D of the coclass theory.