Maximum Order of Finite Abelian Subgroups in the Outer Automorphism Group of a Rank n Free Group

Let Fn be a free group of rank n. Denote by Out Fn its outer automorphism group, that is, its automorphism group modulo its inner automorphism group. For arbitrary n, by considering group actions on finite connected graphs, we derived the maximum order of finite abelian subgroups in Out Fn. Moreover, it is shown that the subgroups reaching this maximum order can be determined up to isomorphisms.