The automorphism group of a finitely generated free group is the normal closure of a single element of order 2.

If m < n then a homomorphism Aut(F_n ) to Aut(F_m ) can have image of cardinality at most 2. More generally, this is true of homomorphisms from Aut(F_n ) to any group G that does not contain an isomorphic image of the symmetric group S_n+1.

Strong restricitons are also obtained on maps to groups G which do not contain a copy of W_n (the semidirect product of (Z/2)ⁿ and S_n ), or of a free abelian group of rank n - 1.

These results place constraints on how Aut(F_n ) can act. For example, if n > 2 then Aut(F_n ) has no non-trivial action on the cirlce (by homeomorphisms).