‎In the classical group theory there is‎ an open question‎: ‎Is every torsion free n-Engel group (for n ≥ 4)‎, nilpotent?‎. ‎To answer the question‎, ‎Traustason‎ [11] showed that with some additional conditions all‎ ‎4-Engel groups are locally nilpotent‎. ‎Here‎, ‎we gave some partial‎ answer to this question on Engel fuzzy subgroups‎. ‎We show that if μ is a normal 4-Engel fuzzy‎ subgroup of ...

We consider equational uniication and matching problems where the equational theory contains a nilpotent function, i.e., a function f satisfying f(x;x) = 0 where 0 is a constant. Nilpotent matching and uniication are shown to be NP-complete. In the presence of associativity and commutativity, the problems still remain NP-complete. But when 0 is also assumed to be the unity for the function f, t...

τ(w)(x) = 〈x(w), w〉, w ∈ W, x ∈ g ⊆ End(W ), and similarly for τ ′. Our main theorem describes the behaviour of closures of nilpotent orbits under the action of moment maps. It is easy to see that for a nilpotent coadjoint orbit O ⊆ g∗ the set τ ′(τ−1(O)) is the union of nilpotent coadjoint orbits in g′. It turns out that it is a closure of a single orbit: Theorem 1.1 Let O ⊆ g∗ be a nilpotent ...

Let P be a fixed set of primes, G the category of all groups and grouphomomorphisms, and N the full subcategory of nilpotent groups. In [9], an idempotent functor e: N → N , called P -localization, was defined so as to extend the Z-module-theoretic localization of abelian groups. There are two wellknown extensions of e to G, namely, Bousfield’s P -localization [2], [4], denoted by EZP , and Rib...

Two groups are said to have the same nilpotent genus if they have the same nilpotent quotients. We answer four questions of Baumslag concerning nilpotent completions. (i) There exists a pair of finitely generated, residually torsion-free-nilpotent groups of the same nilpotent genus such that one is finitely presented and the other is not. (ii) There exists a pair of finitely presented, residual...

Let G be a finite group and p an odd prime. By M < G we denote that M is a proper subgroup of G. Put the set Ψ p (G) = {M:M < G, |G : M| 6= a prime power and |G : M|p = 1}. In this paper we investigate the structure of G if every member of Ψ p (G) is nilpotent.

A zero-nonzero pattern A is said to be potentially nilpotent over a field F if there exists a nilpotent matrix with entries in F having zero-nonzero pattern A. We explore the construction of potentially nilpotent patterns over a field. We present classes of patterns which are potentially nilpotent over a field F if and only if the field F contains certain roots of unity. We then introduce some ...