Automorphism groups of free metabelian nilpotent groups

For positive integers n and c, let Fn be the free group of rank n, Fn,c the free nilpotent group of class c and rank n, and Mn,c the free metabelian and nilpotent-of-class-c group of rank n. In this thesis, we give a set of generators of Aut (F n,c) and a set of generators of Aut (Mn,c) for any positive integers n and c. Although these generating sets are not minimal, they are small to some extent and are adequate for presentations. We also give a minimal generating set of Aut ( F2,4) which is used to give a presentation in the text.;For c ≤ 4, a two generator free nilpotent group of class c is also metabelian. For c ≥ 5, a two generator free nilpotent group of class c is not metabelian. We give presentations of Aut (F2,c) for c ≤ 4. We also give a presentation of Aut ( M2,5), which illustrates the algorithm for finding presentations developed in the thesis.;An automorphism of a group G is called an IA-automorphism if it induces the identity automorphism of G/ G'. Let IA(G) be the group of all IA-automorphisms of G and Inn(G) the group of all inner automorphisms of G. We give a presentation of IA(M2,c)/Inn(M2,c) for any positive integer c.;Using Fox's free partial derivative and the Jacobian matrix, we give a criterion for a set of elements to be a generating set of a quotient group Fn/N, where Fn is the free group of rank n and N is a normal subgroup of Fn. As an application of this criterion, we give necessary and sufficient conditions for a set of elements of the Burnside group B(n,p) of exponent p and rank n to be a generating set.