Induced Automorphisms of Free Groups and Free Metabelian Groups

In realizing the significance of Theorem 4, one should bear in mind that Theorem 2 tells us that a large number of automorphisms of <1> />4 are not induced by automorphisms of <b.Also it is not known at present whether every automorphism of O is induced by an automorphism of F.Theorem 1 is proved by invoking a very important theorem due to J. Nielsen and W. Magnus, which we now proceed to describe.Let us call an automorphism of G which induces the identity automorphism in the abelianized group an IAautomorphism of G.The fact that every automorphism of GG2 is induced by an automorphism of G makes it possible, at least in questions concerned with induced automorphisms, to confine attention to the /^-automorphisms.Now in the case where the rank of G is two, the situation is very pleasant owing to the fact that the only M-automorphisms of G are inner automorphisms (J.Nielsen [11] and S. Bachmuth [2]).In the case where the rank of G is larger than two, the situation is much more complex.Here we have the following theorem due to J. Nielsen [12] and W. Magnus [6]: Suppose F is freely generated by ay,a2,---,aq iq = 3).Then the //1-automorphism group of F is generated by the following automorphisms : a -> aiaalaj1 af1, i^j^l^ i, k