Characterizing a free group in its automorphism group (Q752179)

It is well known that the automorphism group A(F) of a free group F of finite rank \(n\geq 2\) is complete, a result proved by \textit{J. Dyer} and the author [J. Lond. Math. Soc., II. Ser. 11, 181-190 (1975; Zbl 0313.20021)]. The proof consists of proving that the group of inner automorphisms of F is a characteristic subgroup of A(F) which implies the completeness of A(F) by a theorem of Burnside, since F and therefore also A(F) has trivial centre. For the same reason the group I(F) of inner automorphisms of F is isomorphic to F itself with which it is identified. The author proves that not only I(F) is characteristic in A(F) but it is also the only normal subgroup of A(F) which is free of rank n.