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 ...

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.

In the variety of all groups, A. I. Mal’cev [5] proved in 1949 that the free product of two residually torsion-free nilpotent groups is again residually torsion-free nilpotent. This paper is motivated by the analogous question in the variety of metabelian groups: can we determine whether free metabelian products of residually torsion-free nilpotent metabelian groups are residually torsion-free ...

We consider an open problem on the stability of nonlinear nilpotent switched systems posed by Daniel Liberzon. Partial solutions to this problem were obtained as corollaries of global nice reachability results for nilpotent control systems. The global structure is crucial in establishing stability. We show that nice reachability analysis may be reduced to the reachability analysis of a specific...

We study complex product structures on nilpotent Lie algebras, establishing some of their main properties, and then we restrict ourselves to 6 dimensions, obtaining the classification of 6-dimensional nilpotent Lie algebras admitting such structures. We prove that any complex structure which forms part of a complex product structure on a 6-dimensional nilpotent Lie algebra must be nilpotent in ...

Some properties and characterizations for nilpotent matrices are established and in particular, a necessary and sufficient condition for an n × n nilpotent matrix to have the nilpotent index 2 and 3 is given. Mathematics Subject Classification: 15A33; 06D99

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...

This article provides a complete description of the differential Gerstenhaber algebras of all nilpotent complex structures on any real six-dimensional nilpotent algebra. As an application, we classify all pseudo-Kählerian complex structures on six-dimensional nilpotent algebras such that the differential Gerstenhaber algebra of its complex structure is quasi-isomorphic to that of its symplectic...

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...

in which GiCG and Gi+1/Gi ⊂ Z(G/Gi) for all i. We call G solvable if it admits a normal series (1.1) in which Gi+1/Gi is abelian for all i. Every nilpotent group is solvable. Nilpotent groups include finite p-groups, and some theorems about p-groups extend to nilpotent groups (e.g., any nontrivial normal subgroup of a nilpotent group has a nontrivial intersection with the center). There is a la...