Related papers: Weights and Nilpotent Subgroups

Let $B$ be a $p$-block of a finite group, and set $m=$ $\sum \chi(1)^2$, the sum taken over all height zero characters of $B$. Motivated by a result of M. Isaacs characterising $p$-nilpotent finite groups in terms of character degrees, we…

Let $\Gamma_G$ denote a graph associated with a group $G$. A compelling question about finite groups asks whether or not a finite group $H$ must be nilpotent provided $\Gamma_H$ is isomorphic to $\Gamma_G$ for a finite nilpotent group $G$.…

We give effective proofs of residual finiteness and conjugacy separability for finitely generated nilpotent groups. In particular, we give precise asymptotic bounds for a function introduced by Bou-Rabee that measures how large the…

Let $g$ be an element of a group $G$. For a positive integer $n$, let $E_n(g)$ be the subgroup generated by all commutators $[...[[x,g],g],\dots ,g]$ over $x\in G$, where $g$ is repeated $n$ times. We prove that if $G$ is a profinite group…

The parameter coclass has been used successfully in the study of nilpotent algebraic objects of different kinds. In this paper a definition of coclass for nilpotent semigroups is introduced and semigroups of coclass 0, 1, and 2 are…

The so-called "local-global" conjectures in the representation theory of finite groups relate the representation theory of $G$ to that of certain proper subgroups, such as the normalizers of particular $p$-groups. Recent results by several…

Let $\mathcal{N}_{\mathfrak{g}^*}$ be the variety of nilpotent elements in the dual of the Lie algebra of a reductive algebraic group over an algebraically closed field. In \cite{Lu2} Lusztig proposes a definition of a partition of…

We study groups, all maximal nilpotent subgroups of class at most $k$ in which are malnormal. We show that such groups share many similar properties with the ordinary CSA groups. Similarly, we introduce the class of {\em nilpotency…

We show that there exists an algorithm to decide any single equation in the Heisenberg group in finite time. The method works for all two-step nilpotent groups with rank-one commutator, which includes the higher Heisenberg groups. We also…

Let $G$ be a simple algebraic group and $\mathcal O$ a nilpotent orbit in $\mathfrak g$. Let ${\mathbf{CS}}(\mathcal O)$ denote the affine cone over the secant variety of $\overline{\mathbb P\mathcal O}\subset \mathbb P\mathfrak g$. Using…

In this paper we continue the study of powerfully nilpotent groups. These are powerful $p$-groups possessing a central series of a special kind. To each such group one can attach a powerful nilpotency class that leads naturally to the…

This is a survey article with a limited list of references (as required by the publisher) which appears in the Encyclopedia of Mathematical Physics, eds. J.-P. Francoise, G.L. Naber and Tsou S.T. Oxford: Elsevier, 2006. vol.4, pp.94--104.

If $G$ and $H$ are finitely generated residually nilpotent groups, then $G$ and $H$ are in the same nilpotent genus if they have the same lower central quotients (up to isomorphism). A stronger condition is that $H$ is para-$G$ if there…

It is well known that if $G$ is a group and $H$ is a normal subgroup of $G$ of finite index $k$, then $x^k \in H$ for every $x \in G$. We examine finite groups $G$ with the property that $x^k \in H$ for every subgroup $H$ of $G$, where $k$…

Fundamental conjectures in modular representation theory of finite groups, more precisely, Alperin's Weight Conjecture and Robinson's Ordinary Weight Conjecture, can be expressed in terms of fusion systems. We use fusion systems to connect…

We prove that if a linear group $G$ is almost Engel, then $G$ is finite-by-hypercentral. If $G$ is almost nil, then $G$ is finite-by-nilpotent.

We describe an algorithm for deciding whether or not a given finitely generated torsion-free nilpotent group is decomposable as the direct product of nontrivial subgroups.

We define analogues of Verma modules for finite W-algebras. By the usual ideas of highest weight theory, this is a first step towards the classification of finite dimensional irreducible modules. Motivated by known results in type A, we…

We extend results on finite dimensional nilpotent Lie algebras to Leibniz algebras and counterexamples to others are found. One generator algebras are used in these examples and are investigated further.

Let G denote either a special orthogonal group or a symplectic group defined over the complex numbers. We prove the following saturation result for G: given dominant weights \lambda^1, ..., \lambda^r such that the tensor product…