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...

We prove a general theorem for constructing integral quantum cluster algebras over Z[q±1/2], namely, that under mild conditions the forms of nilpotent always possess algebra structures. These are then shown to be isomorphic corresponding upper algebras, again defined Z[q±1/2]. Previously, this was only known acyclic algebras. The is applied that, every symmetrizable Kac–Moody g and Weyl group e...

Let w be a word in the free group of rank n ∈ N and let V(w) be the variety of groups defined by the law w = 1. Define V(w∗) to be the class of all groups G in which for any infinite subsets X1, . . . , Xn there exist xi ∈ Xi, 1 ≤ i ≤ n, such that w(x1, . . . , xn) = 1. Clearly, V(w) ∪ F ⊆ V(w ∗); F being the class of finite groups. In this paper, we investigate some words w and some certain cl...

We associate a graph NG with a group G (called the non-nilpotent graph of G) as follows: take G as the vertex set and two vertices are adjacent if they generate a non-nilpotent subgroup. In this paper we study the graph theoretical properties of NG and its induced subgraph on G\nil(G), where nil(G) = {x ∈ G | 〈x, y〉 is nilpotent for all y ∈ G}. For any finite group G, we prove that NG has eithe...

Abstract A classical result of Baer states that a finite group G which is the product two normal supersoluble subgroups if and only ʹ nilpotent. In this article, we show = AB (respectively, w -supersoluble) B , in permutes with every maximal subgroup each Sylow then -supersoluble), provided We also investigate products defined above when $ A\cap B=1 obtain more general results.

A homogeneous nilpotent Lie group has a scaling automorphism determined by a grading of its Lie algebra. Many proofs of upper bounds for the Dehn function of such a group depend on being able to fill curves with discs compatible with this grading; the area of such discs changes predictably under the scaling automorphism. In this paper, we present combinatorial methods for finding such bounds. U...

Abstract. For a split reductive group scheme Ǧ over a commutative ring k with Weyl group W , there is an important functor Rep(Ǧ, k) → Rep(W, k) defined by taking the zero weight space. We prove that the restriction of this functor to the subcategory of small representations has an alternative geometric description, in terms of the affine Grassmannian and the nilpotent cone of the Langlands dua...

Suppose that V ,W be a symplectic space and a quadratic space respectively over R. Let g1,g2 be the Lie algebras of the groups of isometries of two spaces. Let τi :V ⊗RW → gi be the moment maps and O be the closure of a nilpotent orbit O in gi for i = 1 or 2. In this paper we show that τ3−i (τ −1 i (O )) is a union of at most two closures of nilpotent orbits in g3−i .

‎for a finite group $g$ let $nu(g)$ denote the number of conjugacy classes of non-normal subgroups of $g$‎. ‎the aim of this paper is to classify all the non-nilpotent groups with $nu(g)=3$‎.