scholarly journals On Irreducible Linear Groups of Nonprimary Degree

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-20 ◽  
Author(s):  
A. A. Yadchenko
Keyword(s):  
Prime Number ◽  
Linear Group ◽  
Prime Power ◽  
Linear Groups ◽  
Odd Order ◽  
Complex Linear

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.

scholarly journals On the odd order composition factors of finite linear groups

2020 ◽  
Vol 23 (6) ◽  
pp. 1057-1068
Author(s):  
Alexander Betz ◽  
Max Chao-Haft ◽  
Ting Gong ◽  
Anthony Ter-Saakov ◽  
Yong Yang
Keyword(s):  
Linear Group ◽  
Linear Groups ◽  
Composition Series ◽  
Odd Order ◽  
Finite Linear Group ◽  
Composition Factors ◽  
Finite Linear

AbstractIn 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.

Erdös-Ko-Rado theorem in some linear groups and some projective special linear group

2014 ◽  
Vol 51 (1) ◽  
pp. 83-91
Author(s):  
Milad Ahanjideh ◽  
Neda Ahanjideh
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Prime Number ◽  
Linear Group ◽  
Projective Line ◽  
Special Linear Group ◽  
Linear Groups ◽  
Projective Special Linear Group ◽  
Special Linear ◽  
Derangement Graph

Let V be the 2-dimensional column vector space over a finite field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document} (where q is necessarily a power of a prime number) and let ℙq be the projective line over \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}. In this paper, it is shown that GL2(q), for q ≠ 3, and SL2(q) acting on V − {0} have the strict EKR property and GL2(3) has the EKR property, but it does not have the strict EKR property. Also, we show that GLn(q) acting on \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left( {\mathbb{F}_q } \right)^n - \left\{ 0 \right\}$$ \end{document} has the EKR property and the derangement graph of PSL2(q) acting on ℙq, where q ≡ −1 (mod 4), has a clique of size q + 1.

scholarly journals Holomorphic Cohomology of Complex Analytic Linear Groups

1966 ◽  
Vol 27 (2) ◽  
pp. 531-542 ◽  
Author(s):  
G. Hochschild ◽  
G. D. Mostow
Keyword(s):  
Linear Group ◽  
Structure Theory ◽  
Linear Groups ◽  
Dimensional Representation ◽  
Lie Algebra Cohomology ◽  
Analytic Group ◽  
Finite Dimensional Representation ◽  
Semidirect Products ◽  
Finite Dimensional ◽  
Rational Cohomology

Let G be a complex analytic group, and let A be the representation space of a finite-dimensional complex analytic representation of G. We consider the cohomology for G in A, such as would be obtained in the usual way from the complex of holomorphic cochains for G in A. Actually, we shall use a more conceptual categorical definition, which is equivalent to the explicit one by cochains. In the context of finite-dimensional representation theory, nothing substantial is lost by assuming that G is a linear group. Under this assumption, it is the main purpose of this paper to relate the holomorphic cohomology of G to Lie algebra cohomology, and to the rational cohomology, in the sense of [1], of algebraic hulls of G. This is accomplished by using the known structure theory for complex analytic linear groups in combination with certain easily established results concerning the cohomology of semidirect products. The main results are Theorem 4.1 (whose hypothesis is always satisfied by a complex analytic linear group) and Theorems 5.1 and 5.2. These last two theorems show that the usual abundantly used connections between complex analytic representations of complex analytic groups and rational representations of algebraic groups extend fully to the superstructure of cohomology.

Two-Dimensional Linear Groups Over Local Rings

1969 ◽  
Vol 21 ◽  
pp. 106-135 ◽  
Author(s):  
Norbert H. J. Lacroix
Keyword(s):  
Local Ring ◽  
Linear Group ◽  
General Linear Group ◽  
General Linear ◽  
Linear Groups ◽  
Local Rings ◽  
Two Dimensional ◽  
Normal Subgroups ◽  
Field Case

The problem of classifying the normal subgroups of the general linear group over a field was solved in the general case by Dieudonné (see 2 and 3). If we consider the problem over a ring, it is trivial to see that there will be more normal subgroups than in the field case. Klingenberg (4) has investigated the situation over a local ring and has shown that they are classified by certain congruence groups which are determined by the ideals in the ring.Klingenberg's solution roughly goes as follows. To a given ideal , attach certain congruence groups and . Next, assign a certain ideal (called the order) to a given subgroup G. The main result states that if G is normal with order a, then ≧ G ≧ , that is, G satisfies the so-called ladder relation at ; conversely, if G satisfies the ladder relation at , then G is normal and has order .

On the contractibility to a point of the linear groups of reflexive non-commutative Lp-spaces

1996 ◽  
Vol 119 (3) ◽  
pp. 545-560 ◽  
Author(s):  
Sergei V. Ferleger ◽  
Fyodor A. Sukochev
Keyword(s):  
Banach Space ◽  
Linear Group ◽  
Operator Norm ◽  
Linear Groups ◽  
Lp Spaces ◽  
Continuous Map

For every Banach space X, denote by GL(X) the linear group of X, i.e. the group of all linear continuous invertible operators on X with the topology induced by the operator norm. One says that GL(X) is contractible to a point if there exists a continuous map F: GL(X) × [0, 1] → GL(X) such that F(A,0) = A and F(A, 1) = Id, for every A ∈ GL(X).

scholarly journals Prime Power Representations of Finite Linear Groups II

1957 ◽  
Vol 9 ◽  
pp. 347-351 ◽  
Author(s):  
Robert Steinberg
Keyword(s):  
Lie Groups ◽  
Structural Properties ◽  
Lie Group ◽  
Prime Power ◽  
Linear Groups ◽  
Finite Linear

The aim of this paper is two-fold: first, to extend the results of (4) to the exceptional finite Lie groups recently discovered by Chevalley (1), and, secondly, to give a construction which works simultaneously for the groups An, Bn, Cn, Dn, En, F4 and G2 (in the usual Lie group notation), and which depends only on intrinsic structural properties of these groups.

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