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!