# Derived length gives no upper bound on nilpotency class

## Statement

For $l > 1$, there exist Nilpotent group (?)s of Solvable length (?) $l$ and arbitrarily large Nilpotence class (?).

## Proof

### Dihedral groups

We first show that for $l = 2$, there exist groups of arbitrarily large nilpotence class.

For $n \ge 3$, the dihedral group $D_{2^n}$, given by the presentation:

$\langle a,x \mid a^{2^{n-1}} = x^2 = e, xax = a^{-1} \rangle$,

has nilpotence class $n - 1$, but solvable length $2$, since it has an abelian normal subgroup $\langle a \rangle$ such that the quotient is also an abelian group.

To get an example of a group of length exactly $l$ for $l > 2$ that has arbitrarily large nilpotence class, take the direct product of $D_{2^n}$ with any nilpotent group of solvable length precisely $l$.