scholarly journals Free subgroups in maximal subgroups of skew linear groups

2019 ◽  
Vol 29 (03) ◽  
pp. 603-614 ◽  
Author(s):  
Bui Xuan Hai ◽  
Huynh Viet Khanh
Keyword(s):  
Linear Group ◽  
Division Ring ◽  
General Linear Group ◽  
Subnormal Subgroup ◽  
Free Subgroup ◽  
Maximal Subgroups ◽  
Linear Groups ◽  
Locally Finite ◽  
Starting Point ◽  
Free Subgroups

The study of the existence of free groups in skew linear groups have begun since the last decades of the 20th century. The starting point is the theorem of Tits (1972), now often referred to as Tits’ Alternative, stating that every finitely generated subgroup of the general linear group [Formula: see text] over a field [Formula: see text] either contains a non-cyclic free subgroup or it is solvable-by-finite. In this paper, we study the existence of non-cyclic free subgroups in maximal subgroups of an almost subnormal subgroup of the general skew linear group over a locally finite division ring.

NILPOTENT AND LOCALLY FINITE MAXIMAL SUBGROUPS OF SKEW LINEAR GROUPS

2011 ◽  
Vol 10 (04) ◽  
pp. 615-622 ◽  
Author(s):  
M. RAMEZAN-NASSAB ◽  
D. KIANI
Keyword(s):  
Maximal Subgroup ◽  
Natural Number ◽  
Division Ring ◽  
Subnormal Subgroup ◽  
Maximal Subgroups ◽  
Linear Groups ◽  
Finitely Generated ◽  
Locally Finite ◽  
Matrix Groups ◽  
Nilpotent Maximal Subgroup

Let D be a division ring and N be a subnormal subgroup of D*. In this paper we prove that if M is a nilpotent maximal subgroup of N, then M′ is abelian. If, furthermore every element of M is algebraic over Z(D) and M′ ⊈ F* or M/Z(M) or M′ is finitely generated, then M is abelian. The second main result of this paper concerns the subgroups of matrix groups; assume D is a noncommutative division ring, n is a natural number, N is a subnormal subgroup of GLn(D), and M is a maximal subgroup of N. We show that if M is locally finite over Z(D)*, then M is either absolutely irreducible or abelian.

Free subgroups in maximal subgroups of SLn(D)

2020 ◽  
pp. 2150071
Author(s):  
R. Fallah-Moghaddam
Keyword(s):  
Division Ring ◽  
Finite Simple Group ◽  
Abelian Subgroup ◽  
Subnormal Subgroup ◽  
Free Subgroup ◽  
Maximal Subgroups ◽  
Central Division ◽  
Normal Abelian Subgroup ◽  
Finite Dimensional ◽  
Free Subgroups

Given a non-commutative finite-dimensional [Formula: see text]-central division ring [Formula: see text], [Formula: see text] a subnormal subgroup of [Formula: see text] and [Formula: see text] a non-abelian maximal subgroup of [Formula: see text], then either [Formula: see text] contains a non-cyclic free subgroup or there exists a non-central maximal normal abelian subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] is a subfield of [Formula: see text], [Formula: see text] is Galois and [Formula: see text], also [Formula: see text] is a finite simple group with [Formula: see text].

scholarly journals Irreducible locally nilpotent finitary skew linear groups

1995 ◽  
Vol 38 (1) ◽  
pp. 63-76 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Linear Group ◽  
Division Ring ◽  
Linear Groups ◽  
Locally Finite ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finitary Linear Group ◽  
Locally Nilpotent ◽  
Finitary Linear ◽  
Finite Dimensional Case

Let V be a left vector space over the arbitrary division ring D and G a locally nilpotent group of finitary automorphisms of V (automorphisms g of V such that dimDV(g-1)<∞) such that V is irreducible as D-G bimodule. If V is infinite dimensional we show that such groups are very rare, much rarer than in the finite-dimensional case. For example we show that if dimDV is infinite then dimDV = |G| = ℵ0 and G is a locally finite q-group for some prime q ≠ char D. Moreover G is isomorphic to a finitary linear group over a field. Examples show that infinite-dimensional such groups G do exist. Note also that there exist examples of finite-dimensional such groups G that are not isomorphic to any finitary linear group over a field. Generally the finite-dimensional examples are more varied.

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 .

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

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.

Existence of Nonabelian Free Subgroups in the Maximal Subgroups of GLn(D)

2014 ◽  
Vol 21 (03) ◽  
pp. 483-496 ◽  
Author(s):  
H. R. Dorbidi ◽  
R. Fallah-Moghaddam ◽  
M. Mahdavi-Hezavehi
Keyword(s):  
Maximal Subgroup ◽  
Division Algebra ◽  
Free Subgroup ◽  
Maximal Subgroups ◽  
Central Division ◽  
Finite Dimensional ◽  
Central Division Algebra ◽  
Free Subgroups ◽  
Maximal Subfield

Given a non-commutative finite dimensional F-central division algebra D, we study conditions under which every non-abelian maximal subgroup M of GLn(D) contains a non-cyclic free subgroup. In general, it is shown that either M contains a non-cyclic free subgroup or there exists a unique maximal subfield K of Mn(D) such that NGLn(D)(K*)=M, K* ◁ M, K/F is Galois with Gal (K/F) ≅ M/K*, and F[M]=Mn(D). In particular, when F is global or local, it is proved that if ([D:F], Char (F))=1, then every non-abelian maximal subgroup of GL1(D) contains a non-cyclic free subgroup. Furthermore, it is also shown that GLn(F) contains no solvable maximal subgroups provided that F is local or global and n ≥ 5.

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.

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.

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