Related papers: Regular orbits of quasisimple linear groups II

Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic $p \geqslant 0$, let $H$ be a proper closed subgroup of $G$ and let $V$ be a nontrivial finite dimensional irreducible rational $KG$-module.…

In this paper, we give a characterization of the action of any abelian subgroup G of GL(n, C) on C^n. We prove that any orbit of G is regular with order m<=2n. Moreover, we give a method to determine this order. In the other hand, we…

Let $\hat G$ be the finite simply connected version of an exceptional Chevalley group, and let $V$ be a nontrivial irreducible module, of minimal dimension, for $\hat G$ over its field of definition. We explore the overgroup structure of…

We consider a group $G$ acting on a local dendrite $X$ (in particular on a graph). We give a full characterization of minimal sets of $G$ by showing that any minimal set $M$ of $G$ (whenever $X$ is different from a dendrite) is either a…

Let $G$ be a permutation group on a set $\Omega$. A base for $G$ is a subset of $\Omega$ whose pointwise stabiliser is trivial, and the base size of $G$ is the minimal cardinality of a base. If $G$ has base size $2$, then the corresponding…

If G is a complex semisimple algebraic group, we characterize the normality and the smoothness of its simple linear compactifications, namely those equivariant GxG-compactifications which possess a unique closed orbit and which arise in a…

We initiate the study of subgroups $H$ of the general linear group $GL_{\binom{n}{m}}(R)$ over a commutative ring $R$ that contain the $m$-th exterior power of an elementary group $\bigwedge^mE_n(R)$. Each such group $H$ corresponds to a…

All finite simple groups are determined with the property that every Galois orbit on conjugacy classes has size at most 4. From this we list all finite simple groups $G$ for which the normalized group of central units of the integral group…

We study infinite groups interpretable in power bounded $T$-convex, $V$-minimal or $p$-adically closed fields. We show that if $G$ is an interpretable definably semisimple group (i.e., has no definable infinite normal abelian subgroups)…

Let $G$ be a complex semisimple algebraic group and $X$ be a complex symmetric homogeneous $G$-variety. Assume that both $G$, $X$ as well as the $G$-action on $X$ are defined over real numbers. Then $G(\mathbb{R})$ acts on $X(\mathbb{R})$…

Let H\subset\GL(V) be a connected complex reductive group where V is a finite-dimensional complex vector space. Let v\in V and let G=\{g\in\GL(V)\mid gHv = Hv\}. Following Ra\"is we say that the orbit Hv is \emph{characteristic for H} if…

Let $G$ be a complex connected reductive algebraic group that acts on a smooth complex algebraic variety $X$, and let $E$ be a $G$-equivariant algebraic vector bundle over $X$. A section of $E$ is regular if it is transversal to the zero…

A linear group G on a finite vector space V, (that is, a subgroup of GL(V)) is called (1/2)-transitive if all the G-orbits on the set of nonzero vectors have the same size. We complete the classification of all the (1/2)-transitive linear…

Let $G$ be a finite primitive permutation group on a set $\Omega$ with point stabiliser $H$. Recall that a subset of $\Omega$ is a base for $G$ if its pointwise stabiliser is trivial. We define the base size of $G$, denoted $b(G,H)$, to be…

We study finite transitive permutation groups $G\leqslant\operatorname{Sym}(\Omega)$ such that all orbits of the conjugation action on $G$ of the normaliser of $G$ in $\operatorname{Sym}(\Omega)$ have size bounded by some constant. Our…

This article began as a study of the structure of infinite permutation groups G in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point…

Let $G$ be a finite abelian group. The Erd{\H o}s--Ginzburg--Ziv constant $\mathsf s (G)$ of $G$ is defined as the smallest integer $l \in \mathbb N$ such that every sequence \ $S$ \ over $G$ of length $|S| \ge l$ \ has a zero-sum…

We show that if $H$ is an almost normal subgroup of $G$ such that both $H$ and $G$ are of type $VFP$ and vcd($G$) = vcd($H$) + 1, then $G$ is the fundamental group of a graph of groups in which all vertex and edge groups are commensurable…

A set $B$ is a basis for a vector space $V$ if every element of $V$ can be uniquely written as a linear combination of the elements of $B$. There is a similar definition of a basis for a finite group. We show that certain semidirect…

In this paper we study the orbit closure problem for a reductive group $G\subseteq GL(X)$ acting on a finite dimensional vector space $V$ over $\C$. We assume that the center of $GL(X)$ lies within $G$ and acts on $V$ through a fixed…