Large dimensional classical groups and linear spaces
Alan R. Camina, Nick Gill, A. E. Zalesski
备注：
32 pages. Version 2 has a new format that includes less repetition. It also proves a slightly stronger result; with the addition of our "Concluding Remarks" section the result holds for dimension at least 21
Suppose that a group $G$ has socle $L$ a simple large-rank classical group. Suppose furthermore that $G$ acts transitively on the set of lines of a linear space $\mathcal{S}$. We prove that, provided $L$ has dimension at least 25, then $G$ acts transitively on the set of flags of $\mathcal{S}$ and hence the action is known. For particular families of classical groups our results hold for dimension smaller than 25.
The group theoretic methods used to prove the result (described in Section 3) are robust and general and are likely to have wider application in the study of almost simple groups acting on finite linear spaces.
Using generating functions, we enumerate regular semisimple conjugacy classes in the finite classical groups. For the general linear, unitary, and symplectic groups this gives a different approach to known results; for the special orthogonal groups the results are new.
This paper defines and develops cycle indices for the finite classical groups. These tools are then applied to study properties of a random matrix chosen uniformly from one of these groups. Properties studied by this technique will include semisimplicity, regularity, regular semisimplicity, the characteristic polynomial, number of Jordan blocks, and average order of a matrix.
Motivated by analogous results for the symmetric group and compact Lie groups, we study the distribution of the number of fixed vectors of a random element of a finite classical group. We determine the limiting moments of these distributions, and find exactly how large the rank of the group has to be in order for the moment to stabilize to its limiting value. The proofs require a subtle use of some q-series identities. We also point out connections with orthogonal polynomials.
For each finite classical group $G$, we classify the subgroups of $G$ which act transitively on a $G$-invariant set of subspaces of the natural module, where the subspaces are either totally isotropic or nondegenerate. Our proof uses the classification of the maximal factorisations of almost simple groups. As a first application of these results we classify all point-transitive subgroups of automorphisms of finite thick generalised quadrangles.
It is shown that a wide range of probabilities and limiting probabilities in finite classical groups have integral coefficients when expanded as a power series in 1/q. Moreover it is proved that the coefficients of the limiting probabilities in the general linear and unitary cases are equal modulo 2. The rate of stabilization of the finite dimensional coefficients as the dimension increases is discussed.
For each family of finite classical groups, and their associated simple quotients, we provide an explicit presentation on a specific generating set of size at most 8. Since there exist efficient algorithms to construct this generating set in any copy of the group, our presentations can be used to verify claimed isomorphisms between representations of the classical group. The presentations are available in Magma.
We present a polynomial time Monte-Carlo algorithm for finite simple black box classical groups of odd characteristic which constructs all root ${\rm{SL}}_2(q)$-subgroups associated with the nodes of the extended Dynkin diagram of the corresponding algebraic group.
We estimate the proportion of several classes of elements in finite classical groups which are readily recognised algorithmically, and for which some power has a large fixed point subspace and acts irreducibly on a complement of it. The estimates are used in complexity analyses of new recognition algorithms for finite classical groups in arbitrary characteristic.
In this paper we prove that the finite simple groups $PSp_6(q)$, $Ω_7(q)$ and $PSU_7(q^2)$ are (2,3)-generated for all q. In particular, this result completes the classification of the (2,3)-generated finite classical simple groups up to dimension 7.
Let $H$ be a finite quasisimple classical group, i.e. $H$ is perfect and $S:=H/Z(H)$ is a finite simple classical group. We prove in this paper that, excluding the cases when the simple group $S$ has a very exceptional Schur multiplier such as $\PSL_3(4)$ or $\PSU_4(3)$, $H$ is uniquely determined by the structure of its complex group algebra. The proofs make essential use of the classification of finite simple groups as well as the results on prime power character degrees and relatively small character degrees of quasisimple classical groups.
We introduce a new approach to the study of finite binary permutation groups and, as an application of our method, we prove Cherlin's binary groups conjecture for groups with socle a finite alternating group, and for the $\mathcal{C}_1$-primitive actions of the finite classical groups. Our new approach involves the notion, defined with respect to a group action, of a `\emph{beautiful subset}'. We demonstrate how the presence of such subsets can be used to show that a given action is not binary. In particular, the study of such sets will lead to a resolution of many of the remaining open cases …
This paper describes how to use subgroups to parameterize unipotent classes in the classical algebraic group in characteristic 2. These results can be viewed as an extension of the Bala-Carter Theorem, and give a convenient way to compare unipotent classes in a group $G$ with unipotent classes of a subgroup $X$ where $G$ is exceptional and $X$ is a Levi subgroup of classical type.
A group $G$ is called to be acceptable (due to M. Larsen) if for any finite group $H$, two element-conjugate homomorphisms are globally conjugate. We answer the acceptability question for general linear, special linear, unitary, symplectic and all orthogonal groups over an algebraically closed field of non-zero characteristics.
A classification of maximal subgroups of odd index in finite simple groups was given by Liebeck and Saxl and, independently, Kantor in 1980s. In the cases of alternating groups or classical groups of Lie type over fields of odd characteristics, the classification was not complete. The classification was completed by the author in 2008. In the cases of finite simple classical groups of Lie type we used results obtained in Kleidman's PhD thesis. However there is a number of flaws in this PhD thesis. The flaws were corrected by Bray, Holt, and Roney-Dougal in 2013. In this note we provide a revis…
In this paper, we will show that nonabelian simple classical groups of Lie type are uniquely determined by the structure of their complex group algebras.
University of Szeged（塞格德大学）；HUN-REN–SZTE Analysis and Applications Research Group（匈牙利研究网络-塞格德大学分析与应用研究组）；HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利研究网络阿尔弗雷德·雷尼数学研究所）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学与物理学院）
Let $G$ be a classical algebraic group, $X$ a maximal rank reductive subgroup and $P$ a parabolic subgroup. This paper classifies when $X\G/P$ is finite. Finiteness is proven using geometric arguments about the action of $X$ on subspaces of the natural module for $G$. Infiniteness is proven using a dimension criterion which involves root systems.
Let $G$ be a transitive permutation group of degree $n$ with point stabiliser $H$ and let $r$ be a prime divisor of $n$. We say that $G$ is $r$-elusive if it does not contain a derangement of order $r$. The problem of determining the $r$-elusive primitive groups can be reduced to the almost simple case, and the purpose of this paper is to complete the study of $r$-elusivity for almost simple classical groups. Building on our earlier work for geometric actions of classical groups, in this paper we handle the remaining non-geometric actions where $H$ is almost simple and irreducible. This requir…
In this paper we investigate commuting involution graphs in classical affine Weyl groups. Let $W$ be a classical Weyl group of rank $n$, with $\tilde W$ its corresponding affine Weyl group. Our main result is that if $X$ is a conjugacy class of involutions in $\tilde W$, then the commuting involution graph $\mathcal{C}(\tilde W, X)$ is either disconnected or has diameter at most $n+2$. This bound is known to hold for types $\tilde A_n$ and $\tilde C_n$, so the main work of this paper is to prove the theorem for types $\tilde B_n$ and $\tilde D_n$.
Let $W$ be a vector space over an algebraically closed field $k$. Let $H$ be a quasisimple group of Lie type of characteristic $p\ne {\rm char}(k)$ acting irreducibly on $W$. Suppose also that $G$ is a classical group with natural module $W$, chosen minimally with respect to containing the image of $H$ under the associated representation. We consider the question of when $H$ can act irreducibly on a $G$-constituent of $W^{\otimes e}$ and study its relationship to the maximal subgroup problem for finite classical groups.
The so-called Tits class, associated to an adjoint absolutely almost simple algebraic group, provides a cohomological obstruction for this group to admit an outer automorphism. If the group has inner type, this obstruction is the only one. In this paper, we prove this is not the case for classical groups of outer type, except for groups of type $^2\mathsf{A}_n$ with $n$ even, or $n=5$. More precisely, we prove a descent theorem for exponent $2$ and degree $6$ algebras with unitary involution, which shows that their automorphism groups have outer automorphisms. In all other relevant classical t…
In this paper we study the (2,k)-generation of the finite classical groups SL(4,q), Sp(4,q), SU(4,q^2) and their projective images. Here k is the order of an arbitrary element of SL(2,q), subject to the necessary condition k>= 3. When q is even we allow also k=4.
Let $F$ be a subfield of the algebraic closure of a finite field $\mathbb{F}_p$, $p \ne 2$, and let $R$ denote any ring such that $F[t] \subset R \subsetneq F(t)$. Let $G$ be a classical Chevalley group of adjoint type defined over $R$. We prove that the group $G(R)$ has the $R_{\infty}$-property.
For a prime $r$, we obtain lower bounds on the proportion of $r$-regular elements in classical groups and show that these lower bounds are the best possible lower bounds that do not depend on the order of the defining field. Along the way, we also provide new upper bounds and answer some open questions of the first author, Pálfy and Saxl.
Recently, there has been substantial progress on the Alperin weight conjecture. As a step to establish the Alperin weight conjecture for all finite groups, we prove the inductive blockwise Alperin weight condition for simple groups of classical type under some additional assumption.
In this paper we determine the classical simple groups of dimension r=3,5 which are (2,3)-generated (the cases r = 2, 4 are known). If r = 3, they are PSL_3(q), q <> 4, and PSU_3(q^2), q^2 <> 9, 25. If r = 5 they are PSL_5(q), for all q, and PSU_5(q^2), q^2 >= 9. Also, the soluble group PSU_3(4) is not (2,3)-generated. We give explicit (2,3)-generators of the linear preimages, in the special linear groups, of the (2,3)-generated simple groups.
In constructive recognition of a representation of a Classical group $G$, much attention has been paid to the natural representation as well as to generic (Black Box) algorithms that treat all representations uniformly. There are theoretical and practical improvements to be made by giving special treatment to certain non-natural representations that arise frequently. In this paper we present and analyse a Las Vegas algorithm for rewriting the Symmetric Square representation.
Every commutator preserving bijection of the unipotent radical $Up(2n, R)$ of the Borel subgroup of the classical symplectic group of rank at least 4 over a field $F$ such that $6F=F$ is shown to be the composition of a standard automorphism of $Up(2n,R)$ and a central map. The latter is a bijection which acts as the right multiplication by a matrix from the center of $Up(2n,F)$.
A classical link in 3-space can be represented by a Gauss paragraph encoding a link diagram in a combinatorial way. A Gauss paragraph may code not a classical link diagram, but a diagram with virtual crossings. We present a criterion and a linear algorithm detecting whether a Gauss paragraph encodes a classical link. We describe Wirtinger presentations realizable by virtual link groups.
Associated to every group with a weak spherical Tits system of rank n+1 with an appropriate rank n subgroup, we construct a relative spectral sequence involving group homology of Levi subgroups of both groups. Using the fact that such Levi subgroups frequently split as semidirect products of smaller groups, we prove homological stability results for unitary groups over division rings with infinite centre as well as for special linear and special orthogonal groups over infinite fields.
A $T$-Schottky group is a discrete group of Möbius transformations whose generators identify pairs of, possibly-tangent, Jordan curves on the complex sphere, ${\hat{\IC}}$. If the curves are Euclidean circles then the group is termed classical $T$-Schottky. We describe the boundary of the space of classical $T$-Schottky groups affording two parabolic generators within the larger parameter space of all $T$-Schottky groups with two parabolic generators. This boundary is surprisingly different from that of the larger space. It is analytic while the boundary of the larger space appears to be fract…
In this paper we study the MOR cryptosystem using finite classical Chevalley groups over a finite field of odd characteristic. In the process we develop an algorithm for these Chevalley groups in the same spirit as the row-column operation for special linear group. We focus our study on orthogonal and symplectic groups. We find the hardness of the proposed MOR cryptosystem for these groups.
We prove for finite reductive groups $G$ of classical type, that every irreducible character of $L$ extends to its inertia group in $N$, where $L$ is an abelian centraliser of a Sylow $d$-torus $\mathbf S$ of $G$ and $N:=N_G(\mathbf S)$. This gives a precise description of the irreducible characters of $N$. Furthermore it enables us to verify the McKay conjecture in this situation for $G$ and some primes.
For most classical and similitude groups, we show that each element can be written as a product of two transformations that a) preserve or almost preserve the underlying form and b) whose squares are certain scalar maps. This generalizes work of Wonenburger and Vinroot. As an application, we re-prove and slightly extend a well-known result of Mœglin, Vignéras and Waldspurger on the existence of automorphisms of $p$-adic classical groups that take each irreducible smooth representations to its dual.
We check that the connected centralisers of nilpotent elements in the orthogonal and symplectic groups have Levi decompositions in even characteristic. This provides a justification for the identification of the isomorphism classes of the reductive quotients as stated in [Liebeck, Seitz; Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras].
We prove that if $Γ$ is a lattice in a classical simple Lie group $G$, then the symmetric space of $G$ is $Γ$-equivariantly homotopy equivalent to a proper cocompact $Γ$-CW complex of dimension the virtual cohomological dimension of $Γ$.
We ask if any finite type generalized braid group is a subgroup of some classical Artin braid group. We define a natural map from a given finite type generalized braid group to a classical braid group and ask if this map is an injective homomorphism. We prove that this map is a homomorphism for the braid groups of type A_n, B_n, I_2(k). The injectivity question of this homomorphism (in these particular cases) is not yet settled. If this map is an injective homomorphism then several results will follow. For example it will follow that the Whitehead group, projective class group and the lower K-…
For Classical compact Lie groups, we use Macdonald's formula \cite{Ma} and Ricci curvature for analyzing a "concentration locus", which is a tool to detect where a sequence of metric, Borel measurable spaces concentrates its measure.
We prove an analogue of the celebrated Hall-Higman theorem, which gives a lower bound for the degree of the minimal polynomial of any semisimple element of prime power order $p^{a}$ of a finite classical group in any nontrivial irreducible cross characteristic representation. With a few explicit exceptions, this degree is at least $p^{a-1}(p-1)$.
Let ${\mathbb F}_q$ be a finite field of characteristic two and ${\mathbb F}_q(X_1,...,X_n)$ a rational function field. We use matrix methods to obtain explicit transcendental bases of the invariant subfields of orthogonal groups and pseudo-symplectic groups on ${\mathbb F}_q(X_1,...,X_n)$ over ${\mathbb F}_q$.
We determine the image of the Artin groups of types B and D inside the Iwahori-Hecke algebras, when defined over finite fields, in the semisimple case. This generalizes earlier work on type A by Brunat, Magaard and Marin. In this multi-parameter case, this image depends heavily on the parameters.
The second de Rham cohomology groups of nilpotent orbits in non-compact real forms of classical complex simple Lie algebras are explicitly computed. Furthermore, the first de Rham cohomology groups of nilpotent orbits in non-compact classical simple Lie algebras are computed; they are proven to be zero for nilpotent orbits in all the complex simple Lie algebras. A key component in these computations is a description of the second and first cohomology groups of homogeneous spaces of general connected Lie groups which is obtained here. This description, which generalizes a previous theorem of th…
This is the second part of the works on Hausdorff dimensions of Schottky groups. It has been conjectured that the Hausdorff dimensions of nonclassical Schottky groups are strictly bounded from below. In this second part of our works we provide a resolution of this conjecture, we prove that there exists a universal positive number $λ>0$, such that any finitely-generated non-elementary Kleinian groups with limit set of Hausdorff dimension $<λ$ are classical Schottky groups. We will generalize our previous technologies given in \cite{HS} to prove our general result. Our result can be consider as …
We introduce and develop the model-theoretic notions of absolute connectedness and type-absolute connectedness for groups. We prove that groups of rational points of split semisimple linear groups (that is, Chevalley groups) over arbitrary infinite fields are absolutely connected and characterize connected Lie groups which are type-absolutely connected. We prove that the class of type-absolutely connected group is exactly the class of discretely topologized groups with trivial Bohr compactification, that is the class of minimally almost periodic groups. As an application we generalize some res…
