Having an action between two groups means a map that comply
Then one can assemble a new operation on to construct the semidirect product . The group obtained is by operating
Let be a set and the set of all maps . If we have an action then, we also can give action via
Then we define
the so called permutational wreath product.
This ultra-algebraic construction allow to give a proof of two pillars theorems in group theory: Nielsen – Schreier and Kurosh.
The proof becomes functorial due the properties of this wreath product.
The following diagram is to be exploited