---
tag: MSEHTZY6XZ
kind: statement
role: theorem
name: The normal forms theorem
sources:
  - wikipedia: en:18673708@r1364033949
    locator: Bass–Serre theory, section Basic set-up > The normal forms theorem
    note: adopted
    mention: wikipedia://en/18673708#0P0V
    sections: [Statement]
inventory: [QS511NKYGG]
---

## Statement

Let *g* be an element of *G* = π<sub>1</sub>(**A**, *T*) represented as a product of the form

$$g=a_0e_1a_1\dots e_na_n,$$

where *e*<sub>1</sub>, ..., *e<sub>n</sub>* is a closed edge-path in *A* with the vertex sequence *v*<sub>0</sub>, *v*<sub>1</sub>, ..., *v<sub>n</sub>* = *v*<sub>0</sub> (that is *v*<sub>0</sub>=*o*(*e*<sub>1</sub>), *v<sub>n</sub>* = *t*(*e<sub>n</sub>*) and *v<sub>i</sub>* = *t*(*e<sub>i</sub>*) = *o*(*e*<sub>*i*+1</sub>) for 0 \< *i* \< *n*) and where $a_i\in A_{v_i}$ for *i* = 0, ..., *n*.

Suppose that *g* = 1 in *G*. Then

- either *n* = 0 and *a*<sub>0</sub> = 1 in $A_{v_0}$,
- or *n* > 0 and there is some 0 \< *i* \< *n* such that *e*<sub>*i*+1</sub> = $\overline{e_{i}}$ and $a_i\in \omega_{e_i}(A_{e_i})$.

The normal forms theorem immediately implies that the canonical homomorphisms *A<sub>v</sub>* → π<sub>1</sub>(**A**, *T*) are injective, so that we can think of the vertex groups *A*<sub>*v*</sub> as subgroups of *G*.

Higgins has given a nice version of the normal form using the fundamental groupoid of a graph of groups. This avoids choosing a base point or tree, and has been exploited by Moore.