Related papers: Lie nilpotency indices of modular group algebras

Let $k$ be any positive integer and $G$ a compact (Hausdorff) group. Let $\mf{np}_k(G)$ denote the probability that $k+1$ randomly chosen elements $x_1,\dots,x_{k+1}$ satisfy $[x_1,x_2,\dots,x_{k+1}]=1$. We study the following problem: If…

We classify, up to isomorphism, gradings by abelian groups on nilpotent filiform Lie algebras of nonzero rank. In case of rank 0, we describe conditions to obtain non trivial $\Z_k$-gradings.

In this article, we discuss Lie nilpotency and Lie solvability of non-abelian tensor product of multiplicative Lie algebras. In particular, for giving information concerning the Lie nilpotency (or Lie solvability) of either multiplicative…

In this paper, we systematically investigate the nilpotentizer and nilpotent graph for a Lie superalgebra over the field of characteristic not equal to 2. First, we establish some fundamental properties of the nilpotentizer. Next, we show…

We present a complete list of groups $G$ and fields $F$ for which: (i) the group of normalized units V(FG) of the group algebra FG is locally nilpotent; (ii) the group algebra FG has a finite number of nilpotent elements and V(FG) is an…

We show that every finite group $G$ of size at least $3$ has a nilpotent subgroup of class at most $2$ and size at least $|G|^{1/32\log\log|G|}$. This answers a question of Pyber, and is essentially best possible.

For any grading by an abelian group $G$ on the exceptional simple Lie algebra $\mathcal{L}$ of type $E_6$ or $E_7$ over an algebraically closed field of characteristic zero, we compute the graded Brauer invariants of simple…

The main purpose of this paper is to study the finite-dimensional solvable Lie algebras described in its title, which we call {\em minimal non-${\mathcal N}$}. To facilitate this we investigate solvable Lie algebras of nilpotent length $k$,…

The algebras of the title are infinite-dimensional graded Lie algebras $L= \bigoplus_{i=1}^{\infty}L_i$, over a field of positive characteristic $p$, that are generated by an element of degree $1$ and an element of degree $p$, and satisfy…

A finite dimensional filiform K-Lie algebra is a nilpotent Lie algebra g whose nil index is maximal, that is equal to dim g -1. We describe necessary and sufficient conditions for a filiform algebra over an algebraically closed field of…

Let $G$ be a finite group, and let $V$ be a completely reducible faithful $G$-module. By a result of Glauberman it has been known for a long time that if $G$ is nilpotent of class 2, then $|G| < |V|$. In this paper we generalize this result…

The main result of this paper is to prove that if a (right) Leibniz algebra $L$ is \textit{right nilpotent} of degree $n$, then $L$ is \textit{strongly nilpotent} of degree less or equal to $4n^2-2n+1$.

We classify finite-dimensional nilpotent Lie algebras with $2$-dimensional central commutator ideals admitting a Lie group of automorphisms isomorphic to $SO_2(\mathbb R)$. This enables one to enlarge the class of nilpotent Lie algebras of…

Let G be a profinite group. The following results are proved. The commutator subgroup G' is finite if and only if G is covered by countably many abelian subgroups. The group G is finite-by-nilpotent if and only if G is covered by countably…

In this paper, we investigate nilpotent Lie algebras $ L $ of nilpotency class $3 $ and provide a complete classification of those satisfying $ \dim L^2 = 3 $ and $Z(L) = L^3 \cong A(2). $ Furthermore, we explicitly characterize the…

We prove that if $G$ is finite 2-generated $p$-group of nilpotence class at most 2 then the group algebra of $G$ with coefficients in the field with $p$ elements determines $G$ up to isomorphisms.

Let $p$ be a prime number, let $d$ be an integer and let $G$ be a $d$-generated finite $p$-group of nilpotency class smaller than $p$. Then the number of possible isomorphism types for the mod $p$ cohomology algebra $H^*(G;{\mathbb F}_p)$…

We survey aspects of locally nilpotent linear groups. Then we obtain a new classification; namely, we classify the irreducible maximal locally nilpotent subgroups of $\mathrm{GL}(q, \mathbb F)$ for prime $q$ and any field $\mathbb F$.

We study bounds on nilpotence in H*(BG), the mod p cohomology of the classifying space of a compact Lie group G. Part of this is a report of our previous work on this problem, updated to reflect the consequences of Peter Symonds recent…

Let G be a finite group. The Plesken Lie algebra L[G] is a subalgebra of the complex group algebra C[G] and admits a direct-sum decomposition into simple Lie algebras based on the ordinary character theory of G. In this paper we review the…