scholarly journals Free subgroups in linear groups over some skew fields

1987 ◽  
Vol 105 (1) ◽  
pp. 1-28 ◽  
Author(s):  
A.I Lichtman
Keyword(s):  
Linear Groups ◽  
Skew Fields ◽  
Free Subgroups

scholarly journals Some matrix groups over finite-dimensional division algebras

1990 ◽  
Vol 33 (1) ◽  
pp. 97-111 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Division Algebra ◽  
Finite Dimension ◽  
Linear Groups ◽  
Matrix Groups ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Skew Fields ◽  
Rank 2 ◽  
Soluble Subgroup ◽  
Free Subgroups

Let n be a positive integer and D a division algebra of finite dimension m over its centre. We describe in detail the structure of a soluble subgroup G of GL(n,D). (More generally we consider subgroups of GL{n,D) with no free subgroup of rank 2.) Of course G is isomorphic to a linear group of degree mn and hence linear theory describes G, but the object here is to reduce as far as possible the dependence of the description on m. The results are particularly sharp if n=l. They will be used in later papers to study matrix groups over certain types of infinite-dimensional division algebra. This present paper was very much inspired by A. I. Lichtman's work: Free subgroups in linear groups over some skew fields, J. Algebra105 (1987), 1–28.

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.

On Discrete Free Subgroups of Linear Groups

1978 ◽  
Vol s2-17 (1) ◽  
pp. 79-85 ◽  
Author(s):  
Gerhard Rosenberger
Keyword(s):  
Linear Groups ◽  
Free Subgroups

scholarly journals Free subgroups in almost subnormal subgroups of general skew linear groups

2017 ◽  
Vol 28 (5) ◽  
pp. 707-717 ◽  
Author(s):  
N. K. Ngoc ◽  
M. H. Bien ◽  
B. X. Hai
Keyword(s):  
Linear Groups ◽  
Free Subgroups ◽  
Subnormal Subgroups

scholarly journals Free subgroups in linear groups

1972 ◽  
Vol 20 (2) ◽  
pp. 250-270 ◽  
Author(s):  
J Tits
Keyword(s):  
Linear Groups ◽  
Free Subgroups

scholarly journals Produits de matrices aléatoires et applications aux propriétés géometriques des sous-groupes du groupe linéaire

1990 ◽  
Vol 10 (3) ◽  
pp. 483-512 ◽  
Author(s):  
Yves Guivarc'h
Keyword(s):  
Random Matrices ◽  
Linear Group ◽  
Asymptotic Properties ◽  
Linear Groups ◽  
Products Of Random Matrices ◽  
Free Subgroups

AbstractUsing the asymptotic properties of products of random matrices we study some properties of the subgroups of the linear group. These properties are centered around the theorem of J. Tits giving the existence of free subgroups in linear groups.

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