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

Author(s):  
B. Hartley ◽  
A. E. Zalesskii

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.

1985 ◽  
Vol 37 (2) ◽  
pp. 238-259 ◽  
Author(s):  
John D. Dixon

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


2005 ◽  
Vol 92 (1) ◽  
pp. 62-98 ◽  
Author(s):  
BERND ACKERMANN

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.


Author(s):  
Mai Hoang Bien ◽  
Do Hoang Viet

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.


2009 ◽  
Vol 80 (1) ◽  
pp. 91-104 ◽  
Author(s):  
AZIZOLLAH AZAD ◽  
CHERYL E. PRAEGER

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.


1990 ◽  
Vol 107 (2) ◽  
pp. 193-196 ◽  
Author(s):  
N. A. Vavilov

The purpose of this note is to improve results of J. S. Wilson[12] and L. N. Vaserstein [10] concerning the subnormal structure of the general linear group G = GL (n, R) of degree n ≽ 3 over a commutative ring R. To do this we sharpen results of J. S. Wilson[12], A. Bak[1] and L. N. Vaserstein[10] on subgroups normalized by a relative elementary subgroup. It should be said also that (especially for the case n = 3) our proof is very much simpler than that of[12, 10]. To formulate our results let us recall some notation.


2018 ◽  
Vol 85 (3-4) ◽  
pp. 422
Author(s):  
Meena Sahai ◽  
R. K. Sharma ◽  
Parvesh Kumari

In this article Jordan regular units have been introduced. In particular, it is proved that for n ≥ 2, the general linear group GL(2; F<sub>2<sup>n</sup></sub>) can be generated by Jordan regular units. Further, presentations of GL(2, F<sub>4</sub>); GL(2, F<sub>8</sub>); GL(2, F<sub>16</sub>) and GL(2, F<sub>32</sub>) have been obtained having Jordan regular units as generators.


1998 ◽  
Vol 1 ◽  
pp. 75-108
Author(s):  
D.I. Deriziotis ◽  
C.P. Gotsis

AbstractIn this paper we prove a conjecture due to R. Carter [2], concerning the action of the finite general linear group GLn(q) on a cuspidal module. As an application of this result, we work out the caseGL4(q).


1969 ◽  
Vol 21 ◽  
pp. 106-135 ◽  
Author(s):  
Norbert H. J. Lacroix

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 .


Author(s):  
L. N. Vaserstein

Let A be an associative ring with 1. For any natural number n, let GLnA denote the group of invertible n by n matrices over A, and let EnA be the subgroup generated by all elementary matrices ai, j, where aεA and 1 ≤ i ≡ j ≤ n. For any (two-sided) ideal B of A, let GLnB be the kernel of the canonical homomorphism GLnA→GLn(A/B) and Gn(A, B) the inverse image of the centre of GLn(A/B) (when n > 1, the centre consists of scalar matrices over the centre of the ring A/B). Let EnB denote the subgroup of GLnB generated by its elementary matrices, and let En(A, B) be the normal subgroup of EnA generated by EnB (when n > 2, the group GLn(A, B) is generated by matrices of the form ai, jbi, j(−a)i, j with aA, b in B, i ≡ j, see [7]). In particuler,is the centre of GLnA


Author(s):  
D. G. Arrell ◽  
E. F. Robertson

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.


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