Subgroups Of Linear Groups Research Articles

交换环上线性群中含根子群的极大子群

本文验证了交换环上线性群中三种含根子群的子群的极大性，并讨论了交换环上线性群中含根子群的极大子群的可能类型。 The maximality of three types of subgroups containing root subgroups in linear groups is verified in this paper. And the possible types of maximal subgroups containing root subgroups in linear groups over commutative ring are discussed.

Parabolic Subgroups of Linear Groups and the Vershik-Kerov Group Over 𝔽2

We give a full description of parabolic subgroups of GL n (𝔽2) and present their properties. We show that all such subgroups are determined by a net of ideals. Using this result we prove that analogous statement is true for the Vershik–Kerov group — the group consisting of infinite matrices having finite number of nonzero coefficients under the main diagonal.

On subnormal subgroups of linear groups

We describe the subnormal subgroups of 2-dimensional linear groups over local and full rings in which 2 is invertible, as well as the subnormal subgroups of symplectic groups over local rings in which 2 is invertible.

Automorphisms and normal subgroups of linear groups

In the present paper, the properties of orders of coprime automorphisms of irreducible linear groups are established and sufficient conditions for the existence of a normal Hall π-subgroup in a given π-separable irreducible complex linear group are obtained. Библиография: 11 наэваний.

To the problem of irreducible conjugately dense subgroups of linear groups

To the problem of irreducible conjugately dense subgroups of linear groups

The Lower Garland of Subgroup Lattices in Linear Groups

The lower garland of some lattices of intermediate subgroups in linear groups is described. The results are applied to the case of subgroup lattices in general and special linear groups over a class of rings that contain the group of rational points T of a maximal nonsplit torus in the corresponding algebraic group. It turns out that the lower garlands of these lattices coincide with their intervals consisting of subgroups lying between T and its normalizer. Bibliography: 10 titles.

Free Unit Groups in Group Algebras

Let K[G] denote the group algebra of a finite group G over a field K. If either char K=0 and G is nonabelian, or K is a nonabsolute field of characteristic π>0 and G/Oπ(G) is nonabelian, then it is well known that the group of units U(K[G]) contains a nonabelian free group. For the most part, this follows from the fact that GL2(K) contains such a free subgroup. In this paper, we refine the above result by showing that there are two cyclic subgroups X and Y of G of prime power order, and two units uX∈U(K[X]) and uY∈U(K[Y]), such that 〈uX,uY〉 contains a nonabelian free group. Indeed, we obtain a rather precise description of these units by using an aspect of Tits' theorem on free subgroups in linear groups.

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

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.

Some matrix groups over finite-dimensional division algebras

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.

On Finite Index Subgroups of Linear Groups

On Finite Index Subgroups of Linear Groups

Free subgroups in linear groups over some skew fields

Free subgroups in linear groups over some skew fields

Conjugacy of net subgroups of linear groups

For the full linear group over a matrix-local ring whose quotient by the Jacobson radical is not the field of two elements, we settle the question of the conjugacy ofD -net subgroups (Ref. Zh. Mat., 1977, 2A280). TwoD -net subgroups are conjugate if and only if theD -nets defining them are similar (i.e., can be transformed into each other by a permutation matrix). An analogous result is obtained forD -net subgroups of the symplectic group over a commutative local ring whose residue field contains more than three elements.

Parabolic congruence subgroups in linear groups

Parabolic subgroups are described for the full and special linear groups over a commutative ring R which contain a principal congruence level a, where a is an ideal of R such that R/a is semilocal. It is assumed that R is generated additively by its invertible elements and that the ring identity can be expressed as a sum of two invertible elements.

A note on free subgroups in linear groups

Free subgroups in linear groups have been studied by Tits in [7]. In this note, we make two remarks which extend slightly some of the results in [7]. Recall the beautiful theorem of Jordan on finite linear groups. There is an integer function A.(n) defined on the set of positive integers such that every finite subgroup G of GL(n, C) contains a normal abelian subgroup A with index [G: A] < A(n). The following theorem asserts that [7, Theorem 11 can be put in an analogous form to Jordan’s theorem.

Hypercentral Unipotent Subgroups of Linear Groups

Hypercentral Unipotent Subgroups of Linear Groups

On Discrete Free Subgroups of Linear Groups

On Discrete Free Subgroups of Linear Groups

Characters and subgroups of linear groups

Characters and subgroups of linear groups

Free subgroups in linear groups

(i) H contains no non-abelian free group. (ii) G has a solvable normal subgroup R such that G,‘R is locally .fnite (i.e., every jinite subset generates a$nite subgroup). (iii) G possesses a subgroup G’ of finite index such that if V’ denotes any composition factor of the k[G’]-module V and k’ the endomorphism ring of V’ (i.e., the centralizer of G’ in End,L V’), then k’ is a jield and V’ has a k’-basis with respect to which the matrices representing the elements of G’ are scalar multiples (by elements of k’) f o matrices whose entries are algebraic over the prime field of k.

Normal Sylow subgroups of linear groups

Normal Sylow subgroups of linear groups