The Orbit-Stabilizer Problem for Linear Groups

1985 ◽  
Vol 37 (2) ◽  
pp. 238-259 ◽  
Author(s):  
John D. Dixon
Keyword(s):  
Vector Space ◽  
Linear Group ◽  
General Linear Group ◽  
General Linear ◽  
Linear Groups ◽  
Rational Field ◽  
Finite Set ◽  
Image Position ◽  
Orbit Problem

Let G be a subgroup of the general linear group GL(n, Q) over the rational field Q, and consider its action by right multiplication on the vector space Qn of n-tuples over Q. The present paper investigates the question of how we may constructively determine the orbits and stabilizers of this action for suitable classes of groups. We suppose that G is specified by a finite set {x1, …, xr) of generators, and investigate whether there exist algorithms to solve the two problems:(Orbit Problem) Given u, v ∊ Qn, does there exist x ∊ G such that ux = v; if so, find such an element x as a word in x1, …, xr and their inverses.(Stabilizer Problem) Given u, v ∊ Qn, describe all words in x1, …, xr and their inverses which lie in the stabilizer

The ideal lattices of the complex group rings of finitary special and general linear groups over finite fields

1994 ◽  
Vol 116 (1) ◽  
pp. 7-25 ◽  
Author(s):  
B. Hartley ◽  
A. E. Zalesskii
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
Direct Limit ◽  
General Linear ◽  
Linear Groups ◽  
Group Rings ◽  
General Linear Groups ◽  
Ideal Lattices ◽  
Image Position ◽  
The Ideal

Letqbe a prime power, which will be fixed throughout the paper, letkbe a field, and letbe the field withqelements. LetGn(k)be the general linear groupGL(n, k), andSn(k)the special linear groupSL(n, k). The corresponding groups overwill be denoted simply byGnandSn. We may embedGn(k)inGn+1(k)via the mapForming the direct limit of the resulting system, we obtain thestable general linear groupG∞(k) overk.

On Primitive Solvable Linear Groups

1971 ◽  
Vol 23 (4) ◽  
pp. 679-685 ◽  
Author(s):  
C. R. B. Wright
Keyword(s):  
Vector Space ◽  
Linear Group ◽  
General Linear Group ◽  
Proper Subgroup ◽  
General Linear ◽  
Linear Groups ◽  
Extension Field ◽  
Linear Transformations ◽  
Direct Sum

Let V be a vector space over the field K. A group G of K-linear transformations of V onto itself is primitive in case no proper nontrivial subspace of V is G-invariant and V cannot be written as a direct sum of proper subspaces permuted among themselves by G. Equivalently, G is primitive on V in case G is irreducible and is not induced from a proper subgroup.Suprunenko showed [3, Theorem 12, p. 28] that the n-dimensional general linear group GL(n, K) has a solvable primitive subgroup only if(1) there is a divisor, m, of n such that K has an extension field of degree m containing a primitive p-th root of 1 for each prime p dividing n/m.The main result of this note is the converse fact.

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 .

Infinite dimensional linear groups

1978 ◽  
Vol 78 (3-4) ◽  
pp. 237-240 ◽  
Author(s):  
D. G. Arrell ◽  
E. F. Robertson
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
Normal Structure ◽  
General Linear ◽  
Noetherian Rings ◽  
Linear Groups ◽  
Infinite Dimensional

SynopsisIn this paper we show that some of Bass' results on the normal structure of the stable general linear group can be extended to infinite dimensional linear groups over non-commutative Noetherian rings.

scholarly journals THE LOEWY SERIES OF THE STEINBERG-PIM OF FINITE GENERAL LINEAR GROUPS

2005 ◽  
Vol 92 (1) ◽  
pp. 62-98 ◽  
Author(s):  
BERND ACKERMANN
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
Indecomposable Module ◽  
Defect Group ◽  
General Linear ◽  
Linear Groups ◽  
General Linear Groups ◽  
Projective Indecomposable Module ◽  
Finite General Linear Group

In this paper we calculate the Loewy series of the projective indecomposable module of the unipotent block contained in the Gelfand–Graev module of the finite general linear group in the case of non-describing characteristic and Abelian defect group.

Intersection graphs of general linear groups

2020 ◽  
pp. 2150039
Author(s):  
Mai Hoang Bien ◽  
Do Hoang Viet
Keyword(s):  
Finite Field ◽  
Linear Group ◽  
General Linear Group ◽  
Undirected Graph ◽  
Intersection Graph ◽  
General Linear ◽  
Linear Groups ◽  
Intersection Graphs ◽  
General Linear Groups ◽  
Vertex Set

Let [Formula: see text] be a field and [Formula: see text] the general linear group of degree [Formula: see text] over [Formula: see text]. The intersection graph [Formula: see text] of [Formula: see text] is a simple undirected graph whose vertex set includes all nontrivial proper subgroups of [Formula: see text]. Two vertices [Formula: see text] and [Formula: see text] of [Formula: see text] are adjacent if [Formula: see text] and [Formula: see text]. In this paper, we show that if [Formula: see text] is a finite field containing at least three elements, then the diameter [Formula: see text] is [Formula: see text] or [Formula: see text]. We also classify [Formula: see text] according to [Formula: see text]. In case [Formula: see text] is infinite, we prove that [Formula: see text] is one-ended of diameter 2 and its unique end is thick.

scholarly journals On Generalized Galois Cyclic Orbit Flag Codes

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 217
Author(s):  
Clementa Alonso-González ◽  
Miguel Ángel Navarro-Pérez
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Linear Group ◽  
General Linear Group ◽  
Cyclic Subgroup ◽  
General Linear ◽  
The Relationship

Flag codes that are orbits of a cyclic subgroup of the general linear group acting on flags of a vector space over a finite field, are called cyclic orbit flag codes. In this paper, we present a new contribution to the study of such codes, by focusing this time on the generating flag. More precisely, we examine those ones whose generating flag has at least one subfield among its subspaces. In this situation, two important families arise: the already known Galois flag codes, in case we have just fields, or the generalized Galois flag codes in other case. We investigate the parameters and properties of the latter ones and explore the relationship with their underlying Galois flag code.

The Dimensions of Irreducible Representations of Linear Groups

1970 ◽  
Vol 22 (2) ◽  
pp. 436-448 ◽  
Author(s):  
R. C. King
Keyword(s):  
Symmetric Group ◽  
Linear Group ◽  
Linear Transformation ◽  
General Linear Group ◽  
Irreducible Representations ◽  
Linear Groups ◽  
Young Tableaux ◽  
Tensor Representations ◽  
Image Position ◽  
The Relationship

The theory of the relationship between the symmetric group on a symbols, Σa, and the general linear group in n-dimensions, GL(n), was greatly developed by Weyl [4] who, in this connection, made use of tensor representations of GL(n). The set of mixed tensorsforms the basis of a representation of GL(n) if all the indices may take the values 1, 2, …, n, and if the linear transformationis associated with every non-singular n × n matrix A. The representation is irreducible if the tensors are traceless and if the sets of covariant indices (α)a and contra variant indices (β)b themselves form the bases of irreducible representations (IRs) of Σa and Σb, respectively. These IRs of Σa and Σb may be specified by Young tableaux [μ]a and [v]b in the usual way [4].

scholarly journals MAXIMAL SUBSETS OF PAIRWISE NONCOMMUTING ELEMENTS OF THREE-DIMENSIONAL GENERAL LINEAR GROUPS

2009 ◽  
Vol 80 (1) ◽  
pp. 91-104 ◽  
Author(s):  
AZIZOLLAH AZAD ◽  
CHERYL E. PRAEGER
Keyword(s):  
Linear Group ◽  
Maximal Torus ◽  
General Linear Group ◽  
Three Dimensional ◽  
General Linear ◽  
Linear Groups ◽  
General Linear Groups ◽  
Maximal Subset ◽  
Group A

AbstractLet G be a group. A subset N of G is a set of pairwise noncommuting elements if xy⁄=yx for any two distinct elements x and y in N. If ∣N∣≥∣M∣ for any other set of pairwise noncommuting elements M in G, then N is said to be a maximal subset of pairwise noncommuting elements. In this paper we determine the cardinality of a maximal subset of pairwise noncommuting elements in a three-dimensional general linear group. Moreover, we show how to modify a given maximal subset of pairwise noncommuting elements into another maximal subset of pairwise noncommuting elements that contains a given ‘generating element’ from each maximal torus.

scholarly journals The Maximum Distinguishing Number of a Group

10.37236/1096 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Melody Chan
Keyword(s):  
Vector Space ◽  
Linear Group ◽  
General Linear Group ◽  
Group Element ◽  
General Linear ◽  
Metacyclic Groups ◽  
Distinguishing Number ◽  
Nontrivial Group ◽  
Class C

Let $G$ be a group acting faithfully on a set $X$. The distinguishing number of the action of $G$ on $X$, denoted $D_G(X)$, is the smallest number of colors such that there exists a coloring of $X$ where no nontrivial group element induces a color-preserving permutation of $X$. In this paper, we show that if $G$ is nilpotent of class $c$ or supersolvable of length $c$ then $G$ always acts with distinguishing number at most $c+1$. We obtain that all metacyclic groups act with distinguishing number at most 3; these include all groups of squarefree order. We also prove that the distinguishing number of the action of the general linear group $GL_n(K)$ over a field $K$ on the vector space $K^n$ is 2 if $K$ has at least $n+1$ elements.

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