So I am writing my master thesis and I came across proposition 1.12 on Oleg Bogopolski's book "Introduction to group theory". That proposition states that;
"Let G be a group acting on a connected graph X without inversion of edges. For any subtree T' of the factor graph G\X there exists a subtree T in X such that $p_T:T\to T'$ is an isomorphism".
The proof uses Zorn's lemma on the set of subtrees of X that project injectively into T'. I found similar arguments on a number of books and notes (including Serre's "Trees") and no one actually proves that this set is non-empty.
At first I was misled to say "yes, at least it contains the edges that project onto $T'$" which I believe is wrong and now I will elaborate why I believe that;
Say, an edge e is projected injectively onto T'. Then $ge\in\mathcal{O}(e)$ ($\mathcal{O}$=orbit) for every $g\in G$ and $p(ge)=\mathcal{O}(ge)=\mathcal{O}(e)=p(e)$. Since e is projected injectively, the equation $ge=e$ must hold. So the relations $\mathcal{O}(e)=\{e\}$ and $Stab_G(e)=G$ must also hold. BUT, in most of the cases, the latter equality does NOT hold.
My only guess is that I haven't mastered what p actually does and what "injective" actually means. Thanks in advance and sorry for the long post.