Let G be an arbitrary group. We show that if the Fitting subgroup of G is nilpotent then it is definable. We show also that the class of groups whose Fitting subgroup is nilpotent of class at most n is elementary. We give an example of a group (arbitrary saturated) whose Fitting subgroup is definable but not nilpotent. Similar results for the soluble radical are given.

We develop a general approach to the study of maximal nilpotent subsemigroups of finite semigroups. This approach can be used to recover many known classifications of maximal nilpotent subsemigroups, in particular, for the symmetric inverse semigroup, the symmetric semigroup, and the factor power of the symmetric group. We also apply this approach to obtain a classification of maximal nilpotent...

— Let g be a finite dimensional complex reductive Lie algebra and S(g) its symmetric algebra. The nilpotent bicone of g is the subset of elements (x, y) of g×g whose subspace generated by x and y is contained in the nilpotent cone. The nilpotent bicone is naturally endowed with a scheme structure, as nullvariety of the augmentation ideal of the subalgebra of S(g) ⊗C S(g) generated by the 2-orde...

I show that simple finite vertex algebras are commutative, and that the Lie conformal algebra structure underlying a reduced (= without nilpotent elements) finite vertex algebra is nilpotent.

We construct examples of two-step and three-step nilpotent Lie groups whose automorphism groups are “small” in the sense of either not having a dense orbit for the action on the Lie group, or being nilpotent (the latter being stronger). From the results we also get new examples of compact manifolds covered by two-step simply connected nilpotent Lie groups which do not admit Anosov automorphisms...

‎we describe a new type of presentation that‎, ‎when consistent‎, ‎describes a polycyclic group‎. ‎this presentation is obtained by‎ ‎refining a series of normal subgroups with‎ ‎abelian sections‎. ‎these presentations can be described effectively in‎ ‎computer-algebra-systems like {scshape gap} or ‎{scshape magma}‎. ‎we study these ‎presentations and‎, ‎in particular‎, ‎we obtain consistency c...

In this paper we extend the method of canonical form for congruence bilinear forms to give classification some subclasses 7-dimensional nilpotent Leibniz algebras. Odd-nilpotent algebras are defined as that its even dimensional ideals in lower central series all zero and complex odd-nilpotent with one Leib ideal is obtained by applying aforementioned method.

In this note, I propose the following conjecture : a finite group G is nilpotent if and only if its largest quotient B-group β(G) is nilpotent. I give a proof of this conjecture under the additional assumption that G be solvable. I also show that this conjecture is equivalent to the following : the kernel of restrictions to nilpotent subgroups is a biset-subfunctor of the Burnside functor. AMS ...

In this note we construct a “Kazhdan-Lusztig type” basis in equivariant K-theory of the nilpotent cone of a simple algebraic group G. This basis conjecturally is very close to the basis of this K-group consisting of irreducible bundles on nilpotent orbits. As a consequence we get a natural (conjectural) construction of Lusztig’s bijection between dominant weights and pairs {nilpotent orbit O, i...

applications of hypergroups have mainly appeared in special subclasses. one of the important subclasses is the class of polygroups. in this paper, we study the notions of nilpotent and solvable polygroups by using the notion of heart of polygroups. in particular, we give a necessary and sufficient condition between nilpotent (solvable) polygroups and fundamental groups.