1
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.