Exercise 9, § 3

Let $G$ and $G'$ be graphs and let $\varphi : G'\rightarrow G$ be a graph morphism such that $(G', \varphi )$ is a covering of G.

a) Prove that the graph $\varphi$(Grp(G$'$)) is a subgroupoid of the groupoid Grp(G).

b) Prove that, for every vertex $a$ of $G'$, the group homomorphism $\pi_1(\varphi , a)$ is injective.

c) Suppose that G is connected and fix a vertex $a$ of G. Let K be a subgroup of $\varpi_{G,a}$. Prove that there exists a covering $(H, \psi )$ of the graph G and a vertex $b$ of H such that $\psi (b) =a$ and K is the image of the morphism $\pi_1(\psi , a)$.

d) Prove that a subgroup of a free group is free (see also IV, p. 417, example 2).

e) Let $n$ and $q$ be integers; if F is a free group on $n$ generators and K a subgroup of F of index $q$, prove that K is a free group on $1+q(n-1)$ generators.