We show nilpotency and completeness results for the homotopy automorphisms of a marked n-stage an unstable coalgebra. These objects figure in moduli problem coalgebras. Our theorems extend classical work Dror, Zabrodsky, Maruyama Møller.

Nagata gave a fundamental sufficient condition on group actions on finitely generated commutative algebras for finite generation of the subalgebra of invariants. In this paper we consider groups acting on noncommutative algebras over a field of characteristic zero. We characterize all the T-ideals of the free associative algebra such that the algebra of invariants in the corresponding relativel...

We present an algorithm to solve a system of diagonal polynomial equations over finite fields when the number of variables is greater than some fixed polynomial of the number of equations whose degree depends only on the degree of the polynomial equations. Our algorithm works in time polynomial in the number of equations and the logarithm of the size of the field, whenever the degree of the pol...

We have proved that 4-Engel groups are locally nilpotent. The proof is based upon detailed computations by both hand and machine. Here we elaborate on explicit computer calculations which provided some of the motivation behind the proof. In particular we give details on the hardest coset enumerations now required to show in a direct proof that 4-Engel p-groups are locally finite for 5 ≤ p ≤ 31....

We study the class of Bernstein algebras that are algebraic, in sense each element generates a finite-dimensional subalgebra. Every algebra has maximal algebraic ideal, and quotient is zero-multiplication algebra. Several equivalent conditions for to be given. In particular, known characterizations train terms nilpotency generalized case locally algebras. Along way, we show if Banach (respectiv...

IN this paper we shall study the connection between local and global nilpotency for groups which satisfy certain chain conditions. In the first section, we shall prove that a group satisfying the chain condition on centralizers has a unique maximal normal nilpotent subgroup (the Fitting subgroup), i.e. the group generated by all normal nilpotent subgroups must again be nilpotent. This generaliz...