# Group theory What do we know about (the structure of) these nilpotent groups?

Leave $$G$$ be the free pro-p group with $$n$$ generators, we can assume $$n = 2$$ and the generators are $$x, y$$ First.
Leave $$G_0 = G$$ Y $$G_ {n + 1} =[G,G_n]$$ Be the group generated by certain switches. Then the quotient group $$Delta_n = G / G_n$$ is a nilpotente group of class $$n$$.

Now I want to know more about the structure of the groups. $$Delta_n$$. For example, $$Delta_1 = mathbb {Z} _px bigoplus mathbb {Z} _py$$ Y $$Delta_2$$ is isomorphic to the group of the upper triangular matrix with diagonal equal to 1 and coefficients belonging to $$mathbb {Z} _p$$.

Also, I suppose $$Delta_ {n + 1} / Delta_n$$ (it's abelian) can be seen as a free $$mathbb {Z} _p$$-module, right? (If so, then the structure of the associated classified Lie algebra is known C.f.Serre "Lie Algebras And Lie Groups").

Any idea or reference will be welcome. Thank you!