# Consequences of Serre's property FA

Proposition 21 of Serre's Trees:

Let G be a group with property FA. If G is contained in an amalgam then G is contained in a conjugate of one of the amalgam's factors.

Can anybody help with this proof?

The amalgam acts on a simplicial tree without inversions. Restricting the action of $G$ to this tree, we have, by property FA, that $G$ fixes a vertex. The stabilizers of vertices are conjugate to the factors of the amalgam. Studying the standard example of $A*_C B$ acting on its Bass-Serre tree will help in understanding why this last statement is true.