Related papers: Regular orbits of sporadic simple groups

Given a finite group $G$ and a faithful irreducible $FG$-module $V$ where $F$ has prime order, does $G$ have a regular orbit on $V$? This problem is equivalent to determining which primitive permutation groups of affine type have a base of…

Let $V$ be a faithful $G$-module for a finite group $G$ and let $p$ be a prime dividing $|G|$. An orbit $v^G$ for the action of $G$ on $V$ is $p$-regular if $|v^G|_p=|G:\bC_G(v)|_p=|G|_p$. Zhang asks the following question in \cite{Zhang}.…

Suppose that a finite solvable group $G$ acts faithfully, irreducibly and quasi-primitively on a finite vector space $V$, and $G$ is not metacyclic. Then $G$ always has a regular orbit on $V$ except for a few "small" cases. We completely…

Let $V$ be a finite-dimensional vector space over a finite field, and suppose $G \leq \Gamma \mathrm{L}(V)$ is a group with a unique subnormal quasisimple subgroup $E(G)$ that is absolutely irreducible on $V$. A base for $G$ is a set of…

Let $G \leq \mathrm{GL}(V)$ be a group with a unique subnormal quasisimple subgroup $E(G)$ that acts absolutely irreducibly on $V$. A base for $G$ acting on $V$ is a set of vectors with trivial pointwise stabiliser in $G$. In this paper we…

Suppose that a finite solvable group $G$ acts faithfully, irreducibly and quasi-primitively on a finite vector space $V$. Then $G$ has a uniquely determined normal subgroup $E$ which is a direct product of extraspecial $p$-groups for…

Let $G$ be a finite group, and let $V$ be a completely reducible faithful $G$-module. It has been known for a long time that if $G$ is abelian, then $G$ has a regular orbit on $V$. In this paper we show that $G$ has an orbit of size at…

A $k$-tuple $(H_1, \ldots, H_k)$ of core-free subgroups of a finite group $G$ is said to be regular if $G$ has a regular orbit on the Cartesian product $G/H_1 \times \cdots \times G/H_k$. The regularity number of $G$, denoted $R(G)$, is the…

Let G be an exceptional simple algebraic group, and let T be a maximal torus in G. In this paper, for every such G, we find all simple rational G-modules V with the following property: for every vector v in V, the closure of its T-orbit is…

Let T be a maximal torus in a classical linear group G. In this paper we find all simple rational G-modules V such that for each vector v in V the closure of its T-orbit is a normal affine variety. For every other G-module we present a…

Suppose that $G$ is a finite solvable group and $V$ is a finite, faithful and completely reducible $G$-module. Let $N$ be a nilpotent subgroup of $G$, then there exits $v \in V$ such that $|\bC_N(v)| \leq (|N|/p)^{1/p}$, where $p$ is the…

Let $G$ be a simple 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 irreducible $KG$-module, which is $p$-restricted, tensor…

We prove a myriad of results related to the stabilizer in an algebraic group $G$ of a generic vector in a representation $V$ of $G$ over an algebraically closed field $k$. Our results are on the level of group schemes, which carries more…

Let $G$ be a finite solvable permutation group acting faithfully and primitively on a finite set $\Omega$. Let $G_0$ be the stabilizer of a point $\alpha \in \Omega$ The rank of $G$ is defined as the number of orbits of $G_0$ in $\Omega$,…

Let $G\leqslant {\rm Sym}(\Omega)$ be a finite transitive permutation group with point stabiliser $H$. A base for $G$ is a subset of $\Omega$ whose pointwise stabiliser is trivial, and the minimal cardinality of a base is called the base…

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…

Building on the classification of modules for algebraic groups with finitely many orbits on subspaces, we determine all faithful irreducible modules for simple and maximal-semisimple connected algebraic groups that are orthogonal and have…

We prove that a finite coprime linear group G in characteristic p>=(|G|-1)/2 has a regular orbit. This bound on p is best possible. We also give an application to blocks with abelian defect groups.

Let $p$ be a prime and $F$ be a finite field of characteristic $p$. Suppose that $FG$ is the group algebra of the finite $p$-group $G$ over the field $F$. Let $V(FG)$ denote the group of normalized units in $FG$ and let $V_*(FG)$ denote the…

Let G be a simply connected semisimple algebraic group over an algebraically closed field k of characteristic 0 and let V be a rational simple G-module of finite dimension. If G/H \subset P(V) is a spherical orbit and if X is its closure,…