Irreducible Linear Groups Research Articles

On a polynomial bound for the orbital diameter of primitive affine groups

Let VG be a finite primitive affine permutation group, where V is a vector space of dimension d over the prime field F p and G is an irreducible linear group on V. We prove that if p divides |G|, then the diameters of all nondiagonal orbital graphs of VG are at most 9 d 3 . This improves an earlier exponential bound by A. Maróti and the author.

Unisingular subgroups of symplectic groups over [formula omitted]

A linear group is called unisingular if every element of it has eigenvalue 1. In this paper we develop some general machinery for the study of unisingular irreducible linear groups. A motivation for the study of such groups comes from several sources, including algebraic geometry, Galois theory, finite group theory and representation theory. In particular, a certain aspect of the theory of abelian varieties requires the knowledge of unisingular irreducible subgroups of the symplectic groups over the field of two elements, and in this paper we concentrate on this special case of the general problem. A more special but important question is that of the existence of such subgroups in the symplectic groups of particular degrees. We answer this question for almost all degrees 2n<250, specifically, the question remains open only 7 values of n.

Glasner property for linear group actions and their products

A theorem of Glasner from 1979 shows that if Y subset mathbb {T}= mathbb {R}/mathbb {Z} is infinite then for each epsilon > 0 there exists an integer n such that nY is epsilon -dense. This has been extended in various works by showing that certain irreducible linear semigroup actions on mathbb {T}^d also satisfy such a Glasner property where each infinite set (in fact, sufficiently large finite set) will have an epsilon -dense image under some element from the acting semigroup. We improve these works by proving a quantitative Glasner theorem for irreducible linear group actions with Zariski connected Zariski closure. This makes use of recent results on linear random walks on the torus. We also pose a natural question that asks whether the Cartesian product of two actions satisfying the Glasner property also satisfy a Glasner property for infinite subsets which contain no two points on a common vertical or horizontal line. We answer this question affirmatively for many such Glasner actions by providing a new Glasner-type theorem for linear actions that are not irreducible, as well as polynomial versions of such results.

Comparing the order and the minimal number of generators of a finite irreducible linear group

We denote the smallest size of a generating set of a group G by d(G). We prove that d(G)log⁡|G|=O(n2log⁡q) for irreducible subgroups G of GL(n,q), and estimate the associated constants. The result is motivated by attempts to bound the complexity of computing the automorphism groups of various classes of finite groups.

Variants of some of the Brauer-Fowler theorems

Brauer and Fowler noted restrictions on the structure of a finite group G in terms of |CG(t)| for an involution t∈G. We consider variants of these themes. We first note that for an arbitrary finite group G of even order, we have|G|<k(F)|CG(t)|4 for each involution t∈G, where F denotes the Fitting subgroup of G and k(F) denotes the number of conjugacy classes of F. In particular, for such a group G we have[G:F(G)]<|CG(t)|4 for each involution t∈G. This result requires the classification of the finite simple groups.The groups SL(2,2n) illustrate that the above exponent 4 cannot be replaced by any exponent less than 3. We do not know at present whether the exponent 4 can be improved in general, though we note that the exponent 3 suffices when G is almost simple.We are however able to prove that every finite group G of even order contains an involution u with[G:F(G)]<|CG(u)|3. The proof of this fact reduces to proving two residual cases: one in which G is almost simple (where the classification of the finite simple groups is needed) and one when G has a Sylow 2-subgroup of order 2. For the latter result, the classification of finite simple groups is not needed (though the Feit-Thompson odd order theorem is).We also prove a very general result on fixed point spaces of involutions in finite irreducible linear groups which does not make use of the classification of the finite simple groups, and some other results on the existence of non-central elements (not necessarily involutions) with large centralizers in general finite groups.Lastly we prove (without using the classification of finite simple groups) that if G is a finite group and t∈G is an involution, then all prime divisors of [G:F(G)] are less than or equal to |CG(t)|+1.

On automorphisms of irreducible linear groups with an Abelian Sylow 2-subgroup

On automorphisms of irreducible linear groups with an Abelian Sylow 2-subgroup

Big elements in irreducible linear groups

Let V be a linear space over a field K of dimension n > 1, and let \({G \leq {\rm GL}(V)}\) be an irreducible linear group. In this paper we prove that the group G contains an element g such that rank \({(g - \alpha E_{n}) \geq \frac{n}{2}}\) for every \({\alpha \in K}\), where En is the identity operator on V. This estimate is sharp for any \({n = 2^{m}}\). The existence of such an element implies that the conjugacy class of G in GL(V) intersects the big Bruhat cell \({B\dot{w}_{0}B}\) of GL(V) non-trivially (here B is a fixed Borel subgroup of G). The latter fact is equivalent to the existence of a complete flag \({\mathfrak{F}}\) such that the flags \({g(\mathfrak{F}), \mathfrak{F}}\) are in general position for some g ∈ G.

Bohr density of simple linear group orbits

AbstractWe show that any non-zero orbit under a non-compact, simple, irreducible linear group is dense in the Bohr compactification of the ambient space.

Hyperoval constructions on the Hermitian surface

New infinite families of hyperovals of the generalized quadrangle H(3,q2) are provided. They arise in different geometric contexts. More precisely, we construct hyperovals by means of certain subsets of the projective plane called here k-tangent arcs with respect to a Hermitian curve (Section 2), hyperovals arising from the geometry of an orthogonal polarity commuting with a unitary polarity (Section 3), hyperovals admitting the irreducible linear group PSL(2,7) as a subgroup of PGU(3,q2), q=ph, p≡3,5or6(mod7) and h an odd integer (Section 4). Finally we construct hyperovals by means of the embedding of PSp(4,q)<PGU(4,q2) as a subfield subgroup (Section 5).

Coprime subdegrees for primitive permutation groups and completely reducible linear groups

In this paper we answer a question of Gabriel Navarro about orbit sizes of a finite linear group H ⊆ GL(V) acting completely reducibly on a vector space V: if the H-orbits containing the vectors a and b have coprime lengths m and n, we prove that the H-orbit containing a + b has length mn. Such groups H are always reducible if n,m > 1. In fact, if H is an irreducible linear group, we show that, for every pair of non-zero vectors, their orbit lengths have a non-trivial common factor.

A problem of Kollár and Larsen on finite linear groups and crepant resolutions

The notion of age of elements of complex linear groups was introduced by M. Reid and is of importance in algebraic geometry, in particular in the study of crepant resolutions and of quotients of Calabi–Yau varieties. In this paper, we solve a problem raised by J. Kollár and M. Larsen on the structure of finite irreducible linear groups generated by elements of age ≤ 1 . More generally, we bound the dimension of finite irreducible linear groups generated by elements of bounded deviation. As a consequence of our main results, we derive some properties of symmetric spaces GU_{d}(\mathbb C)/G having shortest closed geodesics of bounded length, and of quotients \mathbb C^{d}/G having a crepant resolution.

On Irreducible Linear Groups of Nonprimary Degree

It is established that degree 2|A|+1 of irreducible complex linear group with the group A of cosimple automorphisms of odd order is a prime number and proved that if degree 2|H|+1 of π-solvable irreducible complex linear group G with a π-Hall TI-subgroup H is not a prime power, then H is Abelian and normal in G.

Products of conjugacy classes and fixed point spaces

We prove several results on products of conjugacy classes in finite simple groups. The first result is that for any finite nonabelian simple groups, there exists a triple of conjugate elements with product11which generate the group. This result and other ideas are used to solve a 1966 conjecture of Peter Neumann about the existence of elements in an irreducible linear group with small fixed space. We also show that there always exist two conjugacy classes in a finite nonabelian simple group whose product contains every nontrivial element of the group. We use this to show that every element in a nonabelian finite simple group can be written as a product of tworrth powers for any prime powerrr(in particular, a product of two squares answering a conjecture of Larsen, Shalev and Tiep).

Minimal irreducible linear groups and representation of finite groups

Minimal irreducible linear groups and representation of finite groups

Automorphisms and normal subgroups of linear groups

In the present paper, the properties of orders of coprime automorphisms of irreducible linear groups are established and sufficient conditions for the existence of a normal Hall π-subgroup in a given π-separable irreducible complex linear group are obtained. Библиография: 11 наэваний.

Transvection parameter sets in linear groups over associative division rings and special Jordan algebras defined by these rings

In this paper we construct a correspondence between a class of irreducible linear groups of finite degree over an associative division ring D and special Jordan rings which are defined by D.

Irreducible Linear Groups of Degree Four over a Quaternion Division Algebra that Contain a Root Subgroup

Let F be a field of characteristic different from 2, let D be a quaternion division algebra over F and let k be a subfield of F. Assume that D is algebraic over k. We describe the irreducible subgroups of GL 4(D) that contain a conjugate in GL 4(D) of the group consisting of all matrices diag.

Orthogonal linear group–subgroup pairs with the same invariants

The main theorem of Galois theory implies that there are no finite group–subgroup pairs with the same invariants. On the other hand, if we consider linear reductive groups instead of finite groups, the analogous statement is no longer true: There exist counterexample group–subgroup pairs with the same invariants. However, it is possible to classify all these counterexamples for certain types of groups. In [S. Solomon, Irreducible linear group–subgroup pairs with the same invariants, J. Lie Theory 15 (2005), 105–123], we provided the classification for connected complex irreducible groups, and, in this paper, for connected complex reductive orthogonal groups, i.e., groups that preserve some nondegenerate quadratic form.

Irreducible linear groups of degree four over a quaternion division algebra that contain a subgroup [formula omitted

Let F be a field of characteristic ≠2, D a quaternion division algebra over F, K a subfield of D, σ an involutory automorphism of K, Φ 0 a non-degenerate σ-skew-Hermitian form in three variables over K whose Witt index is 1. Suppose that K contains a subfield K 0 of F such that each element of K 0 is invariable under σ and F is algebraic over K 0 . We study the irreducible subgroups of GL 4 ( D ) that contain the subgroup diag ( T 3 ( K , Φ 0 ) , 1 ) .

On the Solvability of Finite Irreducible Linear Groups with Hall TI-Subgroups

Conditions for a π-solvable complex linear group of a relatively small degree to be solvable are found.