Solvable Linear Groups Research Articles

On the odd order subgroups of solvable linear groups

In this paper, we first strengthen a few results about the order of solvable linear groups of odd order. We then use these results to bound the product of the orders of odd-order composition factors in a composition series of an arbitrary finite linear group.

Regular and p-Regular Orbits of Solvable Linear Groups, II

Let [Formula: see text] be a faithful [Formula: see text]-module for a finite group [Formula: see text] and let[Formula: see text] be a prime dividing [Formula: see text]. An orbit [Formula: see text] for the action of [Formula: see text] on[Formula: see text] is regular if [Formula: see text], and is [Formula: see text]-regular if [Formula: see text]. In this note, we study two questions, one by the authors and one by Isaacs, related to the [Formula: see text]-regular orbits and regular orbits of the linear group actions.

Bounding the orders of nilpotent subgroups of solvable linear groups

Bounding the orders of nilpotent subgroups of solvable linear groups

Abelian quotient of subgroups of solvable linear groups

In this paper, we study the relations of the sizes of abelian quotient of a subgroup of finite linear groups and the largest orbit size of the linear group actions. We study two conjectures raised by Keller and the author. We show that these two conjectures are not true in general, but also prove some special cases under the condition that the group is solvable.

Real Liouvillian Extensions of Partial Differential Fields

In this paper, we establish Galois theory for partial differential systems defined over formally real differential fields with a real closed field of constants and over formally p-adic differential fields with a p-adically closed field of constants. For an integrable partial differential system defined over such a field, we prove that there exists a formally real (resp. formally p-adic) Picard-Vessiot extension. Moreover, we obtain a uniqueness result for this Picard-Vessiot extension. We give an adequate definition of the Galois differential group and obtain a Galois fundamental theorem in this setting. We apply the obtained Galois correspondence to characterise formally real Liouvillian extensions of real partial differential fields with a real closed field of constants by means of split solvable linear algebraic groups. We present some examples of real dynamical systems and indicate some possibilities of further development of algebraic methods in real dynamical systems.

Orbit sizes and odd order composition factors of finite linear groups

In this paper, we study the existence of large orbits of odd-order subgroups of a finite solvable linear group G, and present some applications on bounding the odd-order parts of the chief series of an arbitrary linear group G. We also find applications to Gluck's conjecture as well as a related question of Navarro.

On class 2 quotients of linear groups

AbstractIn this paper, we study the relation of the size of the class two quotients of a linear group and the size of the vector space. We answer a question raised in Keller and Yang [Class 2 quotients of solvable linear groups, J. Algebra 509 (2018), 386-396].

On non-Abelian infinite-dimensional linear groups

We study non-Abelian solvable infinite-dimensional linear groups of infinite $p$-rank ($р \geqslant 0$) and of infinite fundamental dimensionality, whose any proper non-Abelian subgroup of infinite $p$-rank has finite fundamental dimensionality. We obtain the description of structure of groups from this class.

Very special algebraic groups

We say that a smooth algebraic group $G$ over a field $k$ is very special if for any field extension $K/k$, every $G_K$-homogeneous $K$-variety has a $K$-rational point. It is known that every split solvable linear algebraic group is very special. In this note, we show that the converse holds, and discuss its relationship with the birational classification of algebraic group actions.

On the odd order composition factors of finite linear groups

Abstract In this paper, we study the product of orders of composition factors of odd order in a composition series of a finite linear group. First we generalize a result by Manz and Wolf about the order of solvable linear groups of odd order. Then we use this result to find bounds for the product of orders of composition factors of odd order in a composition series of a finite linear group.

Class 2 quotients of solvable linear groups

Let G be a finite group, and let V be a completely reducible faithful G-module. By a result of Glauberman it has been known for a long time that if G is nilpotent of class 2, then |G|<|V|. In this paper we generalize this result as follows. Assuming G to be solvable, we show that the order of the maximal class 2 quotient of G is strictly bounded above by |V|.

The large-scale geometry of locally compact solvable groups

This short survey deals with the large-scale geometry of solvable groups. Instead of giving a global overview of this wide subject, we chose to focus on three aspects which illustrate the broad diversity of methods employed in this subject. The first one has probabilistic and analytic flavors, the second is related to cohomological properties of unitary representations, while the third one deals with the Dehn function. To keep the exposition concrete, we discuss lots of examples, mostly among solvable linear groups.

Abelian quotients and orbit sizes of solvable linear groups

Let G be a finite group, and let V be a completely reducible faithful Gmodule. 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 generalize this result as follows. Assuming G to be solvable, we show that G has an orbit of size at least |G/G′| on V. This also strengthens a result of Aschbacher and Guralnick in that situation. Additionally, we prove a similar generalization of the well-known result that if G is nilpotent, then G has an orbit of size at least \(\sqrt {\left| G \right|} \) on V.

A note on real algebraic groups

Abstract The efficacy of using complexifications to understand the structure of real algebraic groups is demonstrated. In particular, the following two results are proved: (i) Let G be a connected solvable linear group whose eigenvalues are all real. If the complexification G ℂ of G is algebraic and operates algebraically on a complex variety V, and some G orbit in V is compact, then this orbit is a point. (ii) If L is a connected subgroup of a connected real linear semisimple group G such that the complexification L ℂ of L is algebraic and L ℂ contains a maximal torus of G ℂ, then L contains a maximal torus of G which complexifies to a maximal torus of G ℂ.

Solvable Groups, Free Divisors and Nonisolated Matrix Singularities I: Towers of Free Divisors

This paper is the first part of a two part paper which introduces a method for determining the vanishing topology of nonisolated matrix singularities. A foundation for this is the introduction in this first part of a method for obtaining new classes of free divisors from representations V of connected solvable linear algebraic groups G. For equidimensional representations where dimG = dimV , with V having an open orbit, we give sufficient conditions that the complement E of this open orbit, the “exceptional orbit variety”, is a free divisor (or a slightly weaker free* divisor). We do so by introducing the notion of a “block representation”which is especially suited for both solvable groups and extensions of reductive groups by them. This is a representation for which the matrix representing a basis of associated vector fields on V defined by the representation can be expressed using a basis for V as a block triangular matrix, with the blocks satisfying certain nonsingularity conditions. We use the Lie algebra structure of G to identify the blocks, the singular structure, and a defining equation for E. This construction naturally fits within the framework of towers of Lie groups and representations yielding a tower of free divisors which allows us to inductively represent the variety of singular matrices as fitting between two free divisors. We specifically apply this to spaces of matrices including m×m symmetric, skew-symmetric or general matrices, where we prove that both the classical Cholesky factorization of matrices and a further “modified Cholesky factorization”which we introduce are given by block representations of solvable group actions. For skew-symmetric matrices, we further introduce an extension of the method valid for a representation of a nonlinear infinite dimensional solvable Lie algebras. In part II, we shall use these geometric decompositions and results for computing the vanishing topology for nonlinear sections of free divisors to compute the vanishing topology for matrix singularities for all of the classes.

Integrality and arithmeticity of solvable linear groups

We develop a practical algorithm to decide whether a finitely generated subgroup of a solvable algebraic group G is arithmetic. This incorporates a procedure to compute a generating set of an arithmetic subgroup of G. We also provide a simple new algorithm for integrality testing of finitely generated solvable-by-finite linear groups over the rational field. The algorithms have been implemented in Magma.

Large orbits of subgroups of solvable linear groups

Suppose that G is a finite solvable group and V is a finite, faithful and completely reducible G-module. Let H be a nilpotent subgroup of G; then H has at least three regular orbits on V ⊕ V. Let H be a subgroup of G and 3 ∤ |H|; then H has at least three regular orbits on V ⊕ V. Let H be a subgroup of G and assume the Sylow 2-subgroups of the semidirect product HV are abelian; then H has at least two regular orbits on V ⊕ V.

Regular and p-regular orbits of solvable linear groups

Let V be a faithful G-module for a finite group G and let p be a prime dividing |G|. An orbit vG for the action of G on V is p-regular if |vG|p=|G:CG(v)|p=|G|p. Zhang asks the following question in Zhang (1993) [8]. Assume that a finite solvable group G acts faithfully and irreducibly on a vector space V over a finite field F. If G has a p-regular orbit for every prime p dividing |G|, is it true that G will have a regular orbit on V? In Lü and Cao (2000) [4], Lü and Cao construct an example showing that the answer to this question is no, however the example itself is not correct. In this paper, we study Zhangʼs question in detail. We construct examples showing that the answer to this question is no in general. We also prove the following result. Assume a finite solvable group G of odd order acts faithfully and irreducibly on a vector space V over a field of odd characteristic. If G has a p-regular orbit for every prime p dividing |G|, then G will have a regular orbit on V.

Sieve methods in group theory I: Powers in linear groups

A general sieve method for groups is formulated. It enables one to “measure” subsets of a finitely generated group. As an application we show that if Γ \Gamma is a finitely generated non-virtually solvable linear group of characteristic zero, then the set of proper powers in Γ \Gamma is exponentially small. This is a far-reaching generalization of a result of Hrushovski, Kropholler, Lubotzky, and Shalev.

Large orbits of odd-order subgroups of solvable linear groups

Suppose that G is a finite solvable group and V is a finite, faithful and completely reducible G-module. Let H be an odd-order subgroup of G, then H has at least two regular orbits on V ⊕ V . Suppose in addition that | V | is odd, then there exists v ∈ V in a regular orbit of F ( G ) ∩ H such that C H ( v ) ⊆ F 2 ( G ) . Let G be a solvable group, H be an odd-order Hall π-subgroup of G, V be a faithful G-module, over possibly different finite fields of odd π-characteristic and assume that V O π ( G ) is completely reducible, then there exists v ∈ V such that C H ( v ) ⊆ O π ( G ) .