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 connected, linear semisimple Lie group with Lie algebra g, and let KC → Aut(pC ) be the complexified isotropy representation at the identity coset of the corresponding symmetric space. The Kostant-Sekiguchi correspondence is a bijection between the nilpotent KC -orbits in pC and the nilpotent G-orbits in g. We show that this correspondence associates each spherical nilpotent KC -orbi...

We describe a new order-theoretic property of the commutator for finite algebras. As a corollary we show that any right nilpotent congruence on a finite algebra is left nilpotent. The result is false for infinite algebras and the converse is false even for finite algebras. We show further that any solvable E-minimal algebra is left nilpotent, any finite algebra whose congruence lattice contains...

In this paper we study degenerations of nilpotent Lie algebras. If λ, μ are two points in the variety of nilpotent Lie algebras, then λ is said to degenerate to μ , λ→deg μ , if μ lies in the Zariski closure of the orbit of λ . It is known that all degenerations of nilpotent Lie algebras of dimension n < 7 can be realized via a one-parameter subgroup. We construct degenerations between characte...

In the present work, we give necessary and sufficient conditions on the graph of a right-angled Artin group that determine whether the group is subgroup separable or not. Also we show that right-angled Artin groups are residually torsion-free nilpotent. Moreover, we investigate the profinite topology of F2 × F2 and of the group L in [18], which are the only obstructions for the subgroup separab...

Let X be a locally finite, vertex-transitive, infinite graph with polynomial growth. Then there exists a quotient group of Aut(X) which contains a finitely generated nilpotent subgroup N which has the same growth rate as X. We show that X contains a subgraph which is finitely contractible onto the h-dimensional lattice, where h is the Hirsch number of N . ∗Supported by FWF (Austria) grant No. P...

We exhibit pseudo Riemannian manifolds which are Szabó nilpotent of arbitrary order, or which are Osserman nilpotent of arbitrary order, or which are Ivanov-Petrova nilpotent of order 3.

 Letbe a ring with an endomorphism and an -derivationAntoine studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. In this paper we introduce and investigate the notion of nil--compatible rings. The class of nil--compatible rings are extended through various ring extensions and many classes of nil--compatible rings are constructed. We al...

Let the group G act transitively on finite set $$\Omega $$ , and let $$S \subseteq G$$ be closed under taking inverses. The Schreier graph $$Sch(G \circlearrowleft \Omega ,S)$$ is with vertex edge $$\{ (\omega ,\omega ^s) : \omega \in s S \}$$ . In this paper, we show that random graphs $$C \log |\Omega |$$ elements exhibit a (two-sided) spectral gap high probability, magnifying well-known theo...

A perfect code in a graph ?=(V,E) is subset C of V such that no two vertices are adjacent and every vertex V?C to exactly one C. subgroup H group G called if there exists Cayley which admits as code. Equivalently, an inverse-closed containing the identity element (A,H) tiling sense can be uniquely expressed product H. In this paper we obtain multiple results on codes finite groups, including fe...