One then checks that for the above action of Aut("F""n") on Outer space "X""n" the subgroup of inner automorphisms Inn("F""n") is contained in the kernel of this action, that is every inner automorphism acts trivially on "X""n". It follows that the action of Aut("F""n") on "X""n" quotients through to an action of Out("F""n") = Aut("F""n")/Inn("F""n") on "X""n". namely, if "φ" ∈ Out("F""n") is an outer automorphism of "F""n" and if "α" in Aut("F""n") is an actual automorphism representing "φ" then for any "x" in "X""n" we have "xφ" = "xα".