Group extensions and graphs

A classical result of Gaschütz affirms that given a finite AA-generated group GG and a prime pp, there exists a group G#G# and an epimorphism φ:G#⟶Gφ:G#⟶G whose kernel is an elementary abelian pp-group which is universal among all groups satisfying this property. This Gaschütz universal extension has also been described in the mathematical literature with the help of the Cayley graph. We give an elementary and self-contained proof of the fact that this description corresponds to the Gaschütz universal extension. Our proof depends on another elementary proof of the Nielsen–Schreier theorem, which states that a subgroup of a free group is free.