A nilpotent group G is a finite group that is the direct product of its Sylow p-subgroups. Theorem 1.1 (Fitting's Theorem) Let G be a finite group, and let H and K be two nilpotent normal subgroups of G. Then HK is nilpotent. Hence in any finite group there is a unique maximal normal nilpotent subgroup, and every nilpotent normal subgroup lies inside this; it is called the Fitting subgroup, and...

in this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. with this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism are defined. n(g) and s(g) are the set of all nilpotency classes and the set of all solvability classes for the group g with respect to different automorphi...

Let R be a commutative ring with unity and M unitary module. Nil(M) the set of all nilpotent elements M. The entire element graph over is an undirected E(G(M)) vertex as any two distinct vertices x y are adjacent if only + ∈ Nil(M). In this paper we attempt to study domination in investigate number well bondage its induced subgraphs N(G(M)) Non(G(M)). Some parameters also studied. It has been s...

Abstract We prove the existence of a limiting distribution for appropriately rescaled diameters random undirected Cayley graphs finite nilpotent groups bounded rank and nilpotency class, thus extending result Shapira Zuck which dealt with case abelian groups. The is defined on space unimodular lattices, as in Our result, when specialised to certain family unitriangular groups, establishes very ...

For any simply connected, simple complex algebraic group, we define upper/lower half-decorated geometric crystals and show that their tropicalization will be normal Kashiwara's crystals. In particular, the of crystal on big Bruhat cell $$ \left(=B{\overline{w}}_0:= {B}^{-}\cap U{\overline{w}}_0U\right) is isomorphic to Langlands dual B(∞) nilpotent-half subalgebra quantum group. As an applicati...

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

In a recent article [Gi99], V.Ginzburg introduced and studied in depth the notion of a principal nilpotent pair in a semisimple Lie algebra g. He also obtained several results for more general pairs. As a next step, we considered in [Pa99] almost principal nilpotent pairs. The aim of this paper is to make a contribution to the general theory of nilpotent pairs. Roughly speaking, a nilpotent pai...

‎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 ...

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 ...

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...