## Automorphism groups of free metabelian nilpotent groups

##### Abstract

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 (' F'2,4) 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 ('F''2,c') for 'c' <= 4. We also give a presentation of Aut (' M'2,5), 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 automorphismof '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.