we call $h$ an $ss$-embedded subgroup of $g$ if there exists a‎ ‎normal subgroup $t$ of $g$ such that $ht$ is subnormal in $g$ and‎ ‎$hcap tleq h_{sg}$‎, ‎where $h_{sg}$ is the maximal $s$-permutable‎ ‎subgroup of $g$ contained in $h$‎. ‎in this paper‎, ‎we investigate the‎ ‎influence of some $ss$-embedded subgroups on the structure of a‎ ‎finite group $g$‎. ‎some new results were obtained.‎

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 consider expansive group actions on a compact metric space containing special fixed point denoted by [Formula: see text], and endomorphisms of such systems whose forward trajectories are attracted toward text]. Such called asymptotically nilpotent, we study the conditions in which they that is, map entire to text] finite number iterations. show for large class discrete groups, this property ...

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...