‎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 give a necessary condition to reduce the Cayley isomorphism problem for Cayley objects of a nilpotent or abelian group G whose order satisfies certain arithmetic properties to the Cayley isomorphism problem of Cayley objects of the Sylow subgroups of G in the case of nilpotent groups, and in the case of abelian groups to certain natural subgroups. As an application of this result, we show th...

Scmicomplctc nilpotcnt groups, that is, nilpotent groups with no outer automorphisms, have been of interest for many years. In this paper pscudocomplete nilpotent groups, that is, nilpotent groups in which the automorphism group and the inner automorphism group arc isomorphic (not equal), are constructed. When suitable conditions are placed on the pseudocomplete nilpotent group, the quotient of...

Let G be a reductive group scheme over the p-adic integers, and let $$\mu $$ minuscule cocharacter for G. In Hodge-type case, we construct functor from nilpotent $$(G,\mu )$$ -displays p-nilpotent rings R to formal p-divisible groups equipped with crystalline Tate tensors. When R/pR has p-basis étale locally, show that this defines an equivalence between two categories. The definition of relies...

Let p be a prime number. We give the explicit structure of 2-nilpotent multiplier for each finite 2-generator p-group class two. Moreover, 2-capable groups in that are characterized.

In the first part, we prove that the dominion (in the sense of Isbell) of a subgroup of a finitely generated nilpotent group is trivial in the category of all nilpotent groups. In the second part, we show that the dominion of a subgroup of a finitely generated nilpotent group of class two is trivial in the category of all metabelian nilpotent groups. Section

Let B be a p-block of a finite group, and set m = ∑ χ(1), the sum taken over all height zero characters of B. Motivated by a result of M. Isaacs characterising p-nilpotent finite groups in terms of character degrees, we show that B is nilpotent if and only if the exact power of p dividing m is equal to the p-part of |G : P ||P : R|, where P is a defect group of B and where R is the focal subgro...

We show that every finitely generated nilpotent group of class 2 occurs as the quotient of a finitely presented abelian-by-nilpotent group by its largest nilpotent normal subgroup.

It is proved that for any prime p a finitely generated nilpotent group is conjugacy separable in the class of finite p-groups if and only if the tor-sion subgroup of it is a finite p-group and the quotient group by the torsion subgroup is abelian. 1. Let K be a class of groups. A group G is called residual K (or K-residual) if for each non-unit element a ∈ G there is a homomorphism ϕ of G onto ...

Motivated by recent results on the minimal base of a permutation group, we introduce new local invariant attached to arbitrary finite groups. More precisely, subset Delta group G is called p-base (where p prime) if generates p-group and C_G(Delta) p-nilpotent. Building Halasi-Mar\'oti, prove that p-solvable groups possess p-bases size 3 for every prime p. For other prominent exhibit 2. In fact,...