Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 104 / 4 / e / Solution

Enumerate $X=\{x_1,\dots ,x_n\}$ and $Y=\{y_1,\dots ,y_n\}$. The universal property of a free group gives an endomorphismSince $Y$ generates, $\alpha$ is surjective. The finitely generated free group is residually finite and hence Hopfian by the preceding part, so $\alpha$ is an automorphism. An automorphism sends a free basis to a free basis; therefore $Y$ is a basis of $F(X)$.