It is well-known that the braid group $B_{n}$ injects into the group of automorphisms of the free group $F_{n+1}$. However, there is certainly a kernel when mapping to the outer automorphism group $Out(F_{n+1})$. Namely, the kernel contains the generator of the center of $B_{n+1}$. Could someone please explain or give a reference to the fact (?) that the whole kernel of $B_{n} \rightarrow Out(F_{n+1})$ is the center of $B_{n}$?