Classifying Finite Simple Groups with Respect to the Number of Orbits Under the Action of the Automorphism Group

Abstract

Let ω(G) denote the number of orbits on the finite group G under the action of Aut(G). Using the classification of finite simple groups, we prove that for any positive integer n, there is only a finite number of (non-abelian) finite simple groups G satisfying ω(G) ≤ n. Then we classify all finite simple groups G such that ω(G) ≤ 17. The latter result was obtained by computational means, using the computer algebra system GAP.     

Remarks on the wonderful compactification of semisimple algebraic groups

We prove that ifG is a semisimple algebraic group of adjoint type over the field of complex numbers,H is the subgroup of all fixed points of an involution σ ofG that is induced by an involution σ of the simply connected coveringĜ ofG, then the wonderful compactification of the homogeneous spaceG/H is isomorphic to the G.I.T quotientG ^ss (L)//H of the wonderful compactificationG ofG for a suitable choice of a line bundleL onG. We also prove a functorial property of the wonderful compactifications of semisimple algebraic groups of adjoint type.     

Frobenius splitting and geometry of G-Schubert varieties

Let X be an equivariant embedding of a connected reductive group G over an algebraically closed field k of positive characteristic. Let B denote a Borel subgroup of G. A G-Schubert variety in X is a subvariety of the form diag(G)⋅V, where V is a B×B-orbit closure in X. In the case where X is the wonderful compactification of a group of adjoint type, the G-Schubert varieties are the closures of Lusztig's G-stable pieces. We prove that X admits a Frobenius splitting which is compatible with all G-Schubert varieties. Moreover, when X is smooth, projective and toroidal, then any G-Schubert variety in X admits a stable Frobenius splitting along an ample divisors. Although this indicates that G-Schubert varieties have nice singularities we present an example of a nonnormal G-Schubert variety in the wonderful compactification of a group of type G₂. Finally we also extend the Frobenius splitting results to the more general class of R-Schubert varieties.     

Projective normality of the wonderful compactification of semisimple adjoint groups

We prove that if G is a semisimple adjoint group over an algebraically closed field of arbitrary characteristic, X is the wonderful compactification of G and is a simply connected covering of G, then for any -linearised very ample line bundle L over X, the cone over X given by L is normal. Received: 1 December 1999; in final form: 19 September 2000 / Published online: 19 October 2001     

Orbits of the Actions of Odd-Order Solvable Groups

Suppose that V is a finite faithful irreducible G-module where G is a finite solvable group of odd order. We prove if the action is quasi-primitive, then either F(G) is abelian or G has at least 212 regular orbits on V. As an application, we prove that when V is a finite faithful completely reducible G-module for a solvable group G of odd order, then there exists v ∈ V such that C _G (v) ? F ₂(G) (where F ₂(G) is the 2nd ascending Fitting subgroup of G). We also generalize a result of Espuelas and Navarro. Let G be a group of odd order and let H be a Hall π-subgroup of G. Let V be a faithful G-module over a finite field of characteristic 2, then there exists v ∈ V such that C _H (v) ? O _π(G).     

The Maximal Chains of the Extended Bruhat Orders on the  × -Orbits of an Infinite Renner Monoid

Let (, S) be a Coxeter system. For ?, δ ? {+, ?} we introduce and investigate combinatorially certain partial orders ≤ _?δ, called extended Bruhat orders, on a  × -set (N, C), which depends on , a subset N ? S, and a component C ? N. We determine the length of the maximal chains between two elements x, y ? (N, C), x ≤ _?δ y.

These posets generalize  equipped with its Bruhat order. They include the  × -orbits of the Renner monoids of reductive algebraic monoids and of some infinite-dimensional generalizations which are equipped with the partial orders obtained by the closure relations of the Bruhat and Birkhoff cells. They also include the  × -orbits of certain posets obtained by generalizing the closure relation of the Bruhat cells of the wonderful compactification.     

The Influence of Hughes Type Action

We call the action of an automorphism α of a finite group G a Hughes type action if it is described by conditions on the orders of elements of G ? α ? ? G. In the present paper we study the structure of finite group G admitting an automorphism α of prime order p so that the orders of elements in G ? α ? ? G are not divisible by p ².     

Regular Action in Rings

Let R be a ring with identity, X(R) the set of all nonzero non-units of R and G(R) the group of all units of R. By considering left and right regular actions of G(R) on X(R), the following are investigated: (1) For a local ring R such that X(R) is a union of n distinct orbits under the left (or right) regular action of G(R) on X(R), if J ⁿ  ≠ 0 = J ⁿ⁺¹ where J is the Jacobson radical of R, then the set of all the distinct ideals of R is exactly {R, J, J ²,…, J ⁿ , 0}, and each orbit under the left regular action is equal to the one under the right regular action. (2) Such a ring R is left (and right) duo ring. (3) For the full matrix ring S of n × n matrices over a commutative ring R, the number of orbits under left regular action of G(S) on X(S) is equal to the number of orbits under right regular action of G(S) on X(S); the result also holds for the ring of n × n upper triangular matrices over R.     

A New Class of Generalized Thompson's Groups and Their Normal Subgroups

For a given group G and a homomorphism ?: G → G × G, we construct groups ?_?(G), 𝒯_?(G), and 𝒱_?(G) that blend Thompson's groups F, T, and V with G, respectively. Furthermore, we describe the lattice of normal subgroups of the groups ?_Δ(G), where Δ: G → G × G is the diagonal homomorphism, Δ(g) = (g, g).     

A Note on the Abelianization Functor

It is well known that the abelianization of a group G can be computed as the cokernel of the diagonal morphism (1_G, 1_G): G → G × G in the category of groups. We generalize this to arbitrary regular subtractive categories, among which are the category of groups, the category of topological groups, and the categories of other group-like structures. We also establish that an abelian category is the same as a regular subtractive category in which every monomorphism is a kernel of some morphism.     

Orbits of strongly solvable spherical subgroups on the flag variety

Let G be a connected reductive complex algebraic group and B a Borel subgroup of G. We consider a subgroup \(H \subset B\) which acts with finitely many orbits on the flag variety G / B, and we classify the H-orbits in G / B in terms of suitable root systems. As well, we study the Weyl group action defined by Knop on the set of H-orbits in G / B, and we give a combinatorial model for this action in terms of weight polytopes.     

On the finiteness of the number of orbits on quasihomogeneous (ℂ*)k × SL 2(ℂ)-manifolds

We obtain an effective criterion for the finiteness of the number of orbits contained in the closure of a given G-orbit for the case of a rational linear action of the group G: = (?*)^k × SL ₂(?) on a finite-dimensional linear space as well as on the projectivization of such a space.     

On Finite Groups with Few Automorphism Orbits

Denote by ω(G) the number of orbits of the action of Aut(G) on the finite group G. We prove that if G is a finite nonsolvable group in which ω(G) ≤5, then G is isomorphic to one of the groups A₅, A₆, PSL(2, 7), or PSL(2, 8). We also consider the case when ω(G) = 6 and show that, if G is a nonsolvable finite group with ω(G) = 6, then either G ≈ PSL(3, 4) or there exists a characteristic elementary abelian 2-subgroup N of G such that G/N ≈ A₅.     

Products of Subgroups Which Are Subgroups

We prove conditions for a product of distinct subgroups of an arbitrary group G to be a subgroup of G. In particular, the normal closure of any A ≤ G is equal to the product of some distinct conjugates of A. As an application of the later result we derive constraints on the size of a nontrivial conjugacy class of a finite non-Abelian simple group.     

Wave Invariants of the Spectrum of the G-Invariant Laplacian and the Basic Spectrum of a Riemannian Foliation

Given a compact boundaryless Riemannian manifold upon which a compact Lie group G acts by isometries, recall that the G-invariant Laplacian is the restriction of the ordinary Laplacian on functions to the space of functions which are constant along the orbits of the action. By considering the wave trace of the invariant Laplacian and the connection between G manifolds and Riemannian foliations, invariants of the spectrum of the G-invariant Laplacian can be computed. These invariants include the lengths of certain geodesic arcs which are orthogonal to the principal orbits and contained in the open dense set of principal orbits are associated to the singularities of the wave trace of the G-invariant spectrum. If the action admits finite orbits, then the invariants also include the lengths of certain geodesics arcs connecting the finite orbit to itself. Under additional hypotheses, we obtain partial wave traces. As an application, a partial trace formula for Riemannian foliations with bundle-like metrics is also presented, as well as several special cases where better results are available.     

Spherical orbit closures in simple projective spaces and their normalizations

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. If G/H ⊂ P(V) is a spherical orbit and if X = [`(G/H)] X = \overline {G/H} is its closure, then we describe the orbits of X and those of its normalization [(X)\tilde] \tilde{X} . If, moreover, the wonderful completion of G/H is strict, then we give necessary and sufficient combinatorial conditions so that the normalization morphism [(X)\tilde] ? X \tilde{X} \to X is a homeomorphism. Such conditions are trivially fulfilled if G is simply laced or if H is a symmetric subgroup.     

Normal Restriction in Finite Groups

A subgroup H of a finite group G is called an NR ? subgroup (Normal Restriction) if whenever K ? H, then K ^G ∩ H = K, where K ^G is the normal closure of K in G. In this article, we will prove some sufficient conditions for the solvability of finite groups which possess many NR-subgroups. We also prove a criterion for the existence of a normal p-complement in finite groups.     

On non-normalizable orbits of isometric actions on Lorentz manifolds

For an isometric action of a Lie group G on a Lorentz manifold (M, g) we consider non-normalizable orbits, i.e. orbits which do not posses a G-invariant normal bundle. Orbits of this type are lightlike. It is shown, that such orbits contain lightlike homogeneous geodesics. Moreover, conditions are given, under which there exists a set of normalizable orbits having an open dense union.     

On closures of orbits and arithmetic of quaternions

ForG=PGL₂(ℚ _p )×PGL₂ ℚ we study the closures of orbits under the maximal split Cartan subgroup ofG in homogeneous spacesΓ\G. We show that if a closure of an orbit contains a closed orbit then the orbit is either dense or closed. We show the relation of this to divisibility properties of integral quaternions and other lattices. Sponsored in part by the Edmund Landau Center for Research in Mathematical Analysis supported by the Minerva Foundation (Germany). Research at MSRI supported by NSF grant DMS8505550.     

The Multiplicity Free Permutation Representations of the Ree Groups 2 G 2(q), the Suzuki Groups 2 B 2(q), and Their Automorphism Groups

Abstract

Let G a simple group of type ² B ₂(q) or ² G ₂(q), where q is an odd power of 2 or 3, respectively. The main goal of this paper is to determine the multiplicity free permutation representations of G and A ≤ Aut(G) where A is a subgroup containing a copy of G. Let B be a Borel subgroup of G. If G = ² B ₂(q) we show that there is only one non-trivial multiplicity free permutation representation, namely the representation of G associated to the action on G/B. If G = ² G ₂(q) we show that there are exactly two such non-trivial representations, namely the representations of G associated to the action on G/B and the action on G/M, where M = UC with U the maximal unipotent subgroup of B and C the unique subgroup of index 2 in the maximal split torus of B. The multiplicity free permutation representations of A correspond to the actions on A/H where H is isomorphic to a subgroup containing B if G = ² B ₂(q), and containing M if G = ² G ₂(q). The problem of determining the multiplicity free representations of the finite simple groups is important, for example, in the classification of distance-transitive graphs.