in this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. with this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism are defined. n(g) and s(g) are the set of all nilpotency classes and the set of all solvability classes for the group g with respect to different automorphi...

Let g be a complex simple Lie algebra, with fixed Borel subalgebra b⊂g and Weyl group W. Expanding on previous work of Fan Stembridge in the simply laced case, this note aims to study fully commutative elements W, their connections spherical nilpotent orbits g. If is not type G2, it shown that an element w∈W if only b determined by inversions w lies closure orbit. A similar characterization als...

τ(w)(x) = 〈x(w), w〉, w ∈ W, x ∈ g ⊆ End(W ), and similarly for τ ′. Our main theorem describes the behaviour of closures of nilpotent orbits under the action of moment maps. It is easy to see that for a nilpotent coadjoint orbit O ⊆ g∗ the set τ ′(τ−1(O)) is the union of nilpotent coadjoint orbits in g′. It turns out that it is a closure of a single orbit: Theorem 1.1 Let O ⊆ g∗ be a nilpotent ...

For a finite group G, let W(G) denotes the set of orders elements G. In this paper we study jW(G)j and show that cyclic order n has maximum value among all groups same order. Furthermore notion in nilpotent non-nilpotent state some inequality for it. Among result minimum is power 2 or it pertains to group.

We give a semi-direct product decomposition of the point stabilisers for the enhanced and exotic nilpotent cones. In particular, we arrive at formulas for the number of points in each orbit over a finite field. This is in accordance with a conjecture of Achar-Henderson. Introduction In the theory of algebraic groups, we find that there is much insight to be gained from studying the nilpotent co...

Let W be a variety of groups defined by a set W of laws and G be a finite p-group in W. The automorphism α of a group G is said to bea marginal automorphism (with respect to W), if for all x ∈ G, x−1α(x) ∈ W∗(G), where W∗(G) is the marginal subgroup of G. Let M,N be two normalsubgroups of G. By AutM(G), we mean the subgroup of Aut(G) consistingof all automorphisms which centralize G/M. AutN(G) ...

We show that if $w$ is a multilinear commutator word and $G$ finite group in which every metanilpotent subgroup generated by $w$-values of rank at most $r$, then the verbal $w(G)$ bounded terms $r$ only. In case where soluble we obtain better result -- nilpotent $r+1$.