  The automorphism group of a finitely generated free group is the normal
closure of a single element of order 2. If $m$ is less than $n$ then a
homomorphism $Aut(F_n)\to Aut(F_m)$ can have cardinality at most 2. More
generally, this is true of homomorphisms from $\Aut(F_n)$ to any group that
does not contain an isomorphic copy of the symmetric group $S_{n+1}$. Strong
constraints are also obtained on maps to groups that do not contain a copy of
$W_n= (\Bbb Z/2)^n\rtimes S_n$, or of $\Bbb Z^{n-1}$. These results place
constraints on how $\Aut(F_n)$ can act. For example, if $n\ge 3$ then any
action of $\Aut(F_n)$ on the circle (by homeomorphisms) factors through
$\text{\rm{det}}:Aut(F_n) \to \Bbb Z_2$ .
