This article examines aspects of the theory of locally nilpotent linear groups. We also present a new classification result for locally nilpotent linear groups over an arbitrary field F. 1. Why Locally Nilpotent Linear Groups? Linear (matrix) groups are a commonly used concrete representation of groups. The first investigations of linear groups were undertaken in the second half of the 19th cen...

let $g$ be a $p$-group of nilpotency class $k$ with finite exponent $exp(g)$ and let $m=lfloorlog_pk floor$. we show that $exp(m^{(c)}(g))$ divides $exp(g)p^{m(k-1)}$, for all $cgeq1$, where $m^{(c)}(g)$ denotes the c-nilpotent multiplier of $g$. this implies that $exp( m(g))$ divides $exp(g)$, for all finite $p$-groups of class at most $p-1$. moreover, we show that our result is an improvement...

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...

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...

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...

The aim of this paper is to prove Theorem 1.1 below, a generalization to virtually nilpotent spaces of a result of Wilkerson [13] and Sullivan [12]. We recall that a CW complex Y is virtually nilpotent if (i) Y is connected, (ii) 7r~ Y is virtually nilpotent (i.e. has a nilpotent subgroup of finite index) and (iii) for every integer n > 1, zr~Y has a subgroup of finite index which acts nilpoten...

In this paper we address the problem of understanding when a verbal subgroup finite group is p -nilpotent, with prime, that is, all its elements ′ -order determine subgroup. We provide two -nilpotency criteria, one for terms lower central series any and derived soluble group, which relies on arithmetic properties related to order products commutators .

Among other things, we prove that the group of automorphisms fixing every normal subgroup of a (nilpotent of class c)-by-abelian group is (nilpotent of class ≤ c)-by-metabelian. In particular, the group of automorphisms fixing every normal subgroup of a metabelian group is soluble of derived length at most 3. An example shows that this bound cannot be improved.