Schur's theorem and its generalisation, Baer's theorem, are distinguished results in group theory, connecting the upper central quotients with lower series. The aim of this paper is to generalise these two different directions, using novel methods related non-abelian tensor product. In particular, we prove a version Schur–Baer for finitely generated groups. Then, apply newly obtained describe k...

The concept of dominion (in the sense of Isbell) is investigated in several varieties of nilpotent groups. A complete description of dominions in the variety of nilpotent groups of class at most 2 is given, and used to prove nontriviality of dominions in the variety of nilpotent groups of class at most c for any c>1 . Some subvarieties of N2 , and the variety of all nilpotent groups of class at...

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 ...

Let g be a complex simple Lie algebra. Fix a Borel subalgebra b and a Cartan subalgebra t ⊂ b. The nilpotent radical of b is denoted by u. The corresponding set of positive (resp. simple) roots is ∆ (resp. Π). An ideal of b is called ad-nilpotent, if it is contained in [b, b]. The theory of ad-nilpotent ideals has attracted much recent attention in the work of Kostant, Cellini-Papi, Sommers, an...

Let c ≥ 0, d ≥ 2 be integers and N (d) c be the variety of groups in which every dgenerator subgroup is nilpotent of class at most c. N.D. Gupta posed this question that for what values of c and d it is true that N (d) c is locally nilpotent? We prove that if c ≤ 2 d + 2 − 3 then the variety N (d) c is locally nilpotent and we reduce the question of Gupta about the periodic groups in N (d) c to...

We compare orbits in the nilpotent cone of type Bn, that of type Cn, and Kato’s exotic nilpotent cone. We prove that the number of Fq-points in each nilpotent orbit of type Bn or Cn equals that in a corresponding union of orbits, called a type-B or type-C piece, in the exotic nilpotent cone. This is a finer version of Lusztig’s result that corresponding special pieces in types Bn and Cn have th...

Recently, the authors obtained Schur multiplier, non-abelian tensor square and exterior of $d$-generator generalized Heisenberg Lie algebras rank $ \frac{1}{2}d(d-1).$ Here, we intend to obtain same results for t$ when \frac{1}{2}d(d-1)-3 \leq t\leq \frac{1}{2}d(d-1)-1.$ Then, as a result, give similar consequences nilpotent algebra L class two \dim (L/Z(L))=d,$ L^2=t such that

Gromov proposed an averaged version of the Dehn function and claimed that in many cases it should be subasymptotic to the Dehn function. Using results on random walks in nilpotent groups, we confirm this claim for most nilpotent groups. In particular, if a nilpotent group satisfies the isoperimetric inequality δ(l) < Clα for α > 2 then it satisfies the averaged isoperimetric inequality δ(l) < C...

Consider analytical three-dimensional differential systems having a singular point at the origin such that its linear part is y∂x−λz∂z for some λ≠0. The restriction of to Center Manifold has nilpotent origin. We prove if restricted system analytic and center origin, with Andreev number 2, then admits formal inverse Jacobi multiplier. also centers systems, on manifolds, are limits Hopf-type cent...