Exercise 8, § 3

Let $G$ be a connected graph and let $a$ be a vertex of G.

a) Let $(G_1, \varphi_1)$ and $(G_2, \varphi_2)$ be coverings of the graph G. Let $u:$ Som(G$_1$)$\rightarrow$ Som(G$_2$) be a mapping such that $u(x)\cdot \gamma$ = $u(x\cdot \gamma )$ for every $\gamma \in \varpi_{G,a}$ and every vertex $x$ of $G_1$ such that Som($\varphi_1$)$(x) =a$. Prove that there exists a unique morphism of graphs $\psi : G_1\rightarrow G_2$ such that $\psi =$ Som($u$).

b) Let X be a set endowed with an operation of the group $\varpi_{G,a}$. Prove that there exists a covering $(G', \varphi )$ of the graph G such that Som($\varphi$ )$^{-1}(a) = X$ and such that the operation of $\varpi_G$ in Som(G$'$) induces the given operation of $\varpi_{G,a}$ in X.