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Is there a constant $K \in \mathbb{N}$ such that for every finite solvable group $G$, there exists a nilpotent subgroup $N \leq G$, and a subset $S \subseteq G$ with $|S| \leq K$, and $\langle N,S\...

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....