---
tag: P5QZQ5GDGX
kind: statement
role: theorem
name: Fundamental theorem of Bass–Serre theory
sources:
  - wikipedia: en:18673708@r1364033949
    locator: Bass–Serre theory, section Fundamental theorem of Bass–Serre theory
    note: adopted
    mention: wikipedia://en/18673708#5OKZ
    sections: [Statement]
inventory: [PH8QVC5PHN]
---

## 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 π<sub>1</sub>(**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* → π<sub>1</sub>(**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.