Among other things, we prove that the group of automorphisms fixing every normal subgroup of a (nilpotent of class c)-by-abelian group is (nilpotent of class ≤ c)-by-metabelian. In particular, the group of automorphisms fixing every normal subgroup of a metabelian group is soluble of derived length at most 3. An example shows that this bound cannot be improved.

In this paper we give an algorithmic characterization of rank two locally nilpotent derivations in dimension three. Together with an algorithm for computing the plinth ideal, this gives a method for computing the rank of a locally nilpotent derivation in dimension three.

We introduce a family of extremal polynomials associated with the prolongation of a stratified nilpotent Lie algebra. These polynomials are related to a new algebraic characterization of abnormal subriemannian geodesics in stratified nilpotent Lie groups. They satisfy a set of remarkable structure relations that are used to integrate the adjoint equations.

We approach the question of which soluble groups are automatic. We describe a class of nilpotent-by-abelian groups which need to be studied in order to answer this question. We show that the nilpotent-by-cyclic groups in this class have exponential isoperimetric inequality and so cannot be automatic.

We describe the nilpotent subgroups of the group Bir(P(C)) of birational transformations of the complex projective plane. Let N be a nilpotent subgroup of class k > 1; then either each element of N has finite order, or N is virtually metabelian.

We investigate some situation in which automorphisms of a groupG are uniquely determined by their restrictions to a proper subgroup H . Much of the paper is devoted to studying under which additional hypotheses this property forces G to be nilpotent if H is. As an application we prove that certain countably infinite locally nilpotent groups have uncountably many (outer) automorphisms.

We prove that in a locally finite variety that has definable principal congruences (DPC), solvable congruences are nilpotent, and strongly solvable congruences are strongly abelian. As a corollary of the arguments we obtain that in a congruence modular variety with DPC, every solvable algebra can be decomposed as a direct product of nilpotent algebras of prime power size.

A general conjecture is given for an explicit basis of the coordinate ring of the closure of the conjugacy class of a nilpotent matrix. This conjecture is proven when the partition given by the transpose Jordan type of the nilpotent matrix is a hook or has two parts.

Finite type nilpotent spaces are weakly equivalent if and only if their singular cochains are quasi-isomorphic as E∞ algebras. The cochain functor from the homotopy category of finite type nilpotent spaces to the homotopy category of E∞ algebras is faithful but not full.

We give a geometric classification of n-dimensional nilpotent, commutative nilpotent and anticommutative algebras. prove that the corresponding varieties are irreducible, find their dimensions describe explicit generic families algebras which define each these varieties. show some applications results in study length