Related papers: 5-Engel Lie algebras II

We prove that 5-Engel Lie algebras over a field of characteristic zero, or over a field of prime characteristic $p>7$, are nilpotent of class at most 11. We also prove that if $G$ is a finite 5-Engel $p$-group for $p>7$ then $G$ is…

The purpose of this paper is to classify all $p$-nilpotent restricted Lie algebras of dimension 5 over perfect fields of characteristic $p\geq 5$. Our work builds upon the recent work of Schneider and Usefi on the classification of…

We continue our study of the Wilson conjecture for $\omega$-categorical Lie algebras and prove that $\omega$-categorical $4$-Engel Lie algebras of characteristic $3$ are nilpotent. We develop a set of tools to adapt in the definable context…

We prove a version of the Wilson conjecture for $\omega$-categorical $3$-Engel Lie algebras over a field of characteristic $5$: every $\omega$-categorical Lie algebra over $\mathbb{F}_5$ which satisfies the identity $[x,y^3] = 0$ is…

The paper presents two new results concerning the varieties of Leibnitz algebras. In the case of prime characteristic p of the base field constructed example not nilpotent variety of Leibnitz algebras satisfying an Engel condition order p.…

In this paper we consider the problem of classifying the $(n-5)$-filiform Lie algebras. This is the first index for which infinite parametrized families appear, as can be seen in dimension $7.$ Moreover we obtain large families of…

In this paper we obtain the classification of $p$-nilpotent restricted Lie algebras of dimension at most four over a perfect field of characteristic p.

Engel subalgebras of finite-dimensional n-Lie algebras are shown to have similar properties to those of Lie algebras. Using these, it is shown that an n-Lie algebra, all of whose maximal subalgebras are ideals, is nilpotent. A primitive…

We prove that the set of right 4-Engel elements of a group $G$ is a subgroup for locally nilpotent groups $G$ without elements of orders 2, 3 or 5; and in this case the normal closure $<x>^G$ is nilpotent of class at most 7 for each right…

The paper studies nilpotent $n$-Lie superalgebras. More specifically speaking, we first prove Engel's theorem for $n$-Lie superalgebras. Second, we research some properties of nilpotent $n$-Lie superalgebras, Finally, we give several…

In this paper we study left 3-Engel elements in groups. In particular, we prove that for any prime $p$ and any left 3-Engel element $x$ of finite $p$-power order in a group $G$, $x^p$ is in the Baer radical of $G$. Also it is proved that…

We study general nilpotent algebras. The results obtained are new even for the classical algebras, such as associative or Lie algebras. We single out certain generic properties of finite-dimensional algebras, mostly over infinite fields.…

Engel subalgebras of finite-dimensional Leibniz algebras are shown to have similar properties to those of Lie algebras. Using these, it is shown that a left Leibniz algebra, all of whose maximal subalgebras are right ideals, is nilpotent. A…

We adopt the $p$-group generation algorithm to classify small-dimensional nilpotent Lie algebras over small fields. Using an implementation of this algorithm, we list the nilpotent Lie algebras of dimension at most~9 over $\F_2$ and those…

In this paper we study left and right 4-Engel elements of a group. In particular, we prove that $<a, a^b>$ is nilpotent of class at most 4, whenever $a$ is any element and $b^{\pm 1}$ are right 4-Engel elements or $a^{\pm 1}$ are left…

Let $P$ be a Poisson algebra with a Lie bracket $\{, \}$ over a field $\F$ of characteristic $p\geq 0$. In this paper, the Lie structure of $P$ is investigated. In particular, if $P$ is solvable with respect to its Lie bracket, then we…

Let R be an associative ring with unity and let [R] and U(R) denote the associated Lie ring (with [a,b]=ab-ba) and the group of units of R, respectively. In 1983 Gupta and Levin proved that if [R] is a nilpotent Lie ring of class c then…

For any odd prime $p$, we give an example of a locally finite $p$-group $G$ containing a left 3-Engel element $x$ where $\langle x \rangle^G$ is not nilpotent.

A Lie algebra L is known to be nilpotent if it admits a grading by (Zp, +) with support X not containing 0. It is also known that the class of L can be bounded by some explicit function of |X|. We generalise this and other classical results…

Let $R_n(G)$ denotes the set of all right $n$-Engel elements of a group $G$. We show that in any group $G$ whose 5th term of lower central series has no element of order 2, $R_3(G)$ is a subgroup. Furthermore we prove that $R_4(G)$ is a…