Fundamental theorem of Bass–Serre theory

Statement

Let G be a group acting on a tree X without inversions. Let A be the quotient graph of groups and let v be a base-vertex in A. Then G is isomorphic to the group π₁(A, v) and there is an equivariant isomorphism between the tree X and the Bass–Serre covering tree $\tilde {\mathbf A}$. More precisely, there is a group isomorphism σ: G → π₁(A, v) and a graph isomorphism $j:X\to \tilde {\mathbf A}$ such that for every g in G, for every vertex x of X and for every edge e of X we have j(gx) = g j(x) and j(ge) = g j(e).

This result is also known as the structure theorem.

One of the immediate consequences is the classic Kurosh subgroup theorem describing the algebraic structure of subgroups of free products.