Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 143 / 1 / a / Solution

The Nielsen–Schreier theorem says that every subgroup of a free group is free. Realize $F_n$ as the fundamental group of the rose $R_n$, the graph with one vertex and $n$ oriented loops. A subgroup $H\le F_n$ of finite index of a subgroup $d$ corresponds to a connected $d$-sheeted covering graph $X\to R_n$. The graph $X$ has $d$ vertices and $dn$ unoriented edges. Choosing a spanning tree leavesedges outside the tree, and these freely generate ${\pi }_1(X)\cong H$. Thus the Nielsen–Schreier formula is