We take a step towards the development of a nilpotency theory for loops based on the commutatorassociator filtration instead of the lower central series. This nilpotency theory shares many essential features with the associative case. In particular, we show that the isolator of the nth commutator-associator subloop coincides with the nth dimension subloop over a field of characteristic zero. Th...

In the present paper we give the full description of the Lie nilpo-tent group algebras which have maximal Lie nilpotency indices.

This article is dedicated to some criteria of generalized nilpotency involving pronormality and abnormality. Also new results on groups, in which abnormality is a transitive relation, have been

We provide a uniform vanishing result for the graded components of finite length Koszul module associated to subspace K⊆∧2V as well sharp upper bound its Hilbert function. This purely algebraic statement has interesting applications study number invariants finitely generated groups, such Alexander invariants, Chen ranks, and degree growth virtual nilpotency class. For instance, we explicitly af...

The purpose of these notes is to give an overview on a class of (graded) Lie algebras, which satisfy certain narrowness conditions on the lattice of their homogeneous ideals. Narrowness conditions arise in the theory of p-groups and pro-p groups. In the group theoretic context, the best known condition for a group is to be of maximal class. Let p be a prime. A group G of order p is said to be a...

In this paper‎, ‎we give a complete proof of Theorem 4.1(ii) and a new‎ ‎elementary proof of Theorem 4.1(i) in [Li and Shen‎, ‎On the‎ ‎intersection of the normalizers of the derived subgroups of all‎ ‎subgroups of a finite group‎, ‎ J‎. ‎Algebra, ‎323  (2010) 1349--1357]‎. ‎In addition‎, ‎we also give a generalization of Baer's Theorem‎.