A bound for the class of nilpotent symplectic alternating algebras

We continue developing the theory of nilpotent symplectic alternating algebras. The algebras of upper bound nilpotent class, that we call maximal algebras, have been introduced and well studied. In this paper, we continue with the external case problem of algebras of minimal nilpotent class. We show the existence of a subclass of algebras over a field [Formula: see text] that is of certain lower bound class that depends on the dimension only. This suggests a minimal bound for the class of nilpotent algebras of dimension [Formula: see text] of rank [Formula: see text] over any field.

The algebraic classification of nilpotent algebras

We give the complete algebraic classification of all complex 4-dimensional nilpotent algebras. The final list has 234 (parametric families of) isomorphism classes of algebras, 66 of which are new in the literature.

Unified computational approach to nilpotent algebra classification problems

In this article, we provide an algorithm with Wolfram Mathematica code that gives a unified computational power in classification of finite dimensional nilpotent algebras using Skjelbred-Sund method. To illustrate the code, we obtain new finite dimensional Moufang algebras.

Factorizations of Finite Groups and Related Topics

This is a survey on the recent progress in the theory of finite groups with factorizations and around it, done by the author and his co-authors, and this has no pretensions to cover all topics in this wide area of research. In particular, we only touch the great consequences of the fundamental paper of Liebeck, Praeger and Saxl on maximal factorizations of almost simple finite groups for the theory of groups with factorizations. In each case the reader can find additional references at the end of Section 1. Some of the methods of investigation can be used to obtain information about finite groups in general, nilpotent algebras and related nearrings.

Congruence preserving expansions of nilpotent algebras

We characterize those algebras that have infinitely many polynomially inequivalent expansions among all finite nilpotent algebras of finite type in congruence modular varieties that are direct products of algebras of prime power order.