Let L and H be finite-dimensional restricted Lie algebras over a perfect field F such that u(L) = u(H), where u(L) is the restricted enveloping algebra of L. We prove that if L is p-nilpotent and abelian, then L = H . We deduce that if L is abelian and F is algebraically closed, then L = H . We use these results to prove the main result of this paper stating that if L is p-nilpotent, then L/L +...

We investigate the properties of Lie algebras of characteristic p which satisfy the Engel identity xy" = 0 for some n < p. We establish a criterion which (when satisfied) implies that if a and b are elements of an Engel-n Lie algebra L then abn~2 generates a nilpotent ideal of I. We show that this criterion is satisfied for n = 6, p = 1, and we deduce that if G is a finite m-generator group of ...

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.

In a recent paper Borel and Serre proved the theorem: If 8 is a Lie algebra of characteristic 0 and 8 has an automorphism of prime period without fixed points f^O, then 8 is nilpotent.1 In this note we give a proof valid also for characteristic p^O. By the same method we can prove several other similar results on automorphisms and derivations. Our method is based on decompositions of the Lie al...

let $g={rm sl}_2(p^f)$ be a special linear group and $p$ be a sylow‎ ‎$2$-subgroup of $g$‎, ‎where $p$ is a prime and $f$ is a positive‎ ‎integer such that $p^f>3$‎. ‎by $n_g(p)$ we denote the normalizer of‎ ‎$p$ in $g$‎. ‎in this paper‎, ‎we show that $n_g(p)$ is nilpotent (or‎ ‎$2$-nilpotent‎, ‎or supersolvable) if and only if $p^{2f}equiv‎ ‎1,({rm mod},16)$‎.

Let P (z) be a formal power series in z = (z1, · · · , zn) with o(P (z)) ≥ 2 and t a formal parameter which commutes with z. We say P (z) is HN (Hessian nilpotent) if its Hessian matrix HesP (z) = ( ∂ 2 P ∂zi∂zj ) is nilpotent. The deformed inversion pair Qt(z) of P (z) by definition is the unique Qt(z) ∈ C[[z, t]] with o(Qt(z)) ≥ 2 such that the formal maps Gt(z) = z + t∇Q(z) and Ft(z) = z − t...

Many work was done for filiform Lie algebras defined by M. Vergne [8]. An interesting fact is that this algebras are obtained by deformations of the filiform Lie algebra Ln,m. This was used for classifications in [4]. Like filiform Lie algebras, filiform Lie superalgebras are obtained by nilpotent deformations of the Lie superalgebra Ln,m. In this paper, we recall this fact and we study even co...

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...