Artin's characterization of the braid subgroup of Aut(F_n)

The image is contained in the set of peripheral-boundary-preserving automorphisms. Let $\beta$ be any braid word. By [F1], $\rho (\beta )(x_i)$ is conjugate to a generator for every $i$ and $\rho (\beta )(x_1\cdots x_n)=x_1\cdots x_n$, so $\rho (\beta )$ is peripheral-boundary-preserving. Hence $\mathrm{im}\rho \subseteq \{$peripheral-boundary-preserving automorphisms$\}$.

The set of peripheral-boundary-preserving automorphisms is contained in the image. Let $A$ be peripheral-boundary-preserving. By [F2] there is a braid word $\beta$ with $A=\rho (\beta )$; hence $A\in \mathrm{im}\rho$.

Uniqueness of the braid. Assume $\rho ({\beta }_1)=\rho ({\beta }_2)$ for braid words ${\beta }_1,{\beta }_2$. Then $\rho ({\beta }_1{\beta }_2^{-1})=\rho ({\beta }_1)\rho ({\beta }_2{)}^{-1}=\mathrm{id}$ because $\rho$ is a homomorphism, so by injectivity [F3] the word ${\beta }_1{\beta }_2^{-1}$ represents the trivial braid, that is, ${\beta }_1={\beta }_2$ in $B_n$. Hence each element of the image is $\rho (\beta )$ for a unique braid $\beta$.

Equality of the two sets. Steps 1.1 and 1.2 give $\mathrm{im}\rho =\{A\in \mathrm{Aut}(F_n):A\text{ is peripheral-boundary-preserving}\}\mathrm{.}$

Conclusion. Step 2.1 identifies the image with the set of peripheral-boundary-preserving automorphisms and step 1.3 shows that the representing braid is unique, which is the characterization of Artin. The necessity and sufficiency directions [F1] and [F2] are choice-free; AC is consumed exactly through the injectivity statement [F3], as declared in the statement. For $n\le 1$ both $F_n$-automorphism conditions are checked directly on the trivial or infinite cyclic group and the same conclusions hold with the trivial braid group.