Locally Finite Normal Subgroups of Absolutely Irreducible Skew Linear Groups

1985 ◽  
Vol s2-32 (1) ◽  
pp. 88-102 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Linear Groups ◽  
Normal Subgroups ◽  
Locally Finite

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 .

scholarly journals Maximal quotient rings of prime group algebras. II Uniform right ideals

1977 ◽  
Vol 24 (3) ◽  
pp. 339-349 ◽  
Author(s):  
John Hannah
Keyword(s):  
Group Algebra ◽  
Quotient Ring ◽  
Group Algebras ◽  
Normal Subgroups ◽  
Locally Finite ◽  
Residually Finite ◽  
Zero Characteristic ◽  
Quotient Rings

AbstractSuppose KG is a prime nonsingular group algebra with uniform right ideals. We show that G has no nontrivial locally finite normal subgroups. If G is soluble or residually finite, or if K has zero characteristic and G is linear, then the maximal right quotient ring of KG is simple Artinian.

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.

scholarly journals An extension of the Kegel–Wielandt theorem to locally finite groups

1996 ◽  
Vol 38 (2) ◽  
pp. 171-176
Author(s):  
Silvana Franciosi ◽  
Francesco de Giovanni ◽  
Yaroslav P. Sysak
Keyword(s):  
Finite Group ◽  
Affirmative Answer ◽  
Locally Finite Group ◽  
Linear Groups ◽  
Arbitrary Group ◽  
Famous Theorem ◽  
Locally Finite ◽  
Locally Nilpotent ◽  
Finite Products ◽  
Open Question

A famous theorem of Kegel and Wielandt states that every finite group which is the product of two nilpotent subgroups is soluble (see [1], Theorem 2.4.3). On the other hand, it is an open question whether an arbitrary group factorized by two nilpotent subgroups satisfies some solubility condition, and only a few partial results are known on this subject. In particular, Kegel [6] obtained an affirmative answer in the case of linear groups, and in the same article he also proved that every locally finite group which is the product of two locally nilpotent FC-subgroups is locally soluble. Recall that a group G is said to be an FC-group if every element of G has only finitely many conjugates. Moreover, Kazarin [5] showed that if the locally finite group G = AB is factorized by an abelian subgroup A and a locally nilpotent subgroup B, then G is locally soluble. The aim of this article is to prove the following extension of the Kegel–Wielandt theorem to locally finite products of hypercentral groups.

Locally soluble skew linear groups

1987 ◽  
Vol 102 (3) ◽  
pp. 421-429 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Group Algebras ◽  
Linear Groups ◽  
Normal Subgroups

In this paper we elucidate the structure of locally soluble absolutely irreducible skew linear groups, and more generally of locally soluble normal subgroups of arbitrary absolutely irreducible skew linear groups. The conclusions are similar to those for soluble such subgroups, but the proofs are not. We also give a couple of applications to the theory of group algebras. Much of our argument works with considerably weaker hypotheses and therefore our methods may well have wider application than just to absolutely irreducible groups.

FINITARY AUTOMORPHISM GROUPS OVER NON-COMMUTATIVE RINGS

2001 ◽  
Vol 64 (3) ◽  
pp. 611-623 ◽  
Author(s):  
B. A. F. WEHRFRITZ
Keyword(s):  
Division Ring ◽  
Automorphism Groups ◽  
Commutative Rings ◽  
Linear Groups ◽  
Locally Finite ◽  
Arbitrary Ring ◽  
Finitary Automorphism

The notion of a group of finitary automorphisms of an arbitrary module over an arbitrary ring is introduced, and it is shown how properties of such groups can be derived from the case where the ring is a division ring (that is, from the properties of finitary skew linear groups). The results are particularly strong if either the group is locally finite or the module is Noetherian.

scholarly journals Injective modules and soluble groups satisfying the minimal condition for normal subgroups

1971 ◽  
Vol 4 (1) ◽  
pp. 113-135 ◽  
Author(s):  
B. Hartley ◽  
D. McDougall
Keyword(s):  
Additive Group ◽  
Nilpotent Groups ◽  
Divisible Group ◽  
Minimal Condition ◽  
Injective Hull ◽  
Normal Subgroups ◽  
Metabelian Groups ◽  
Locally Finite ◽  
Soluble Groups ◽  
Injective Modules

Let p be a prime and let Q be a centre-by-finite p′-group. It is shown that the ZQ-modules which satisfy the minimal condition on submodules and have p–groups as their underlying additive groups can be classified in terms of the irreducible ZpQ-modules. If such a ZQ-module V is indecomposable it is either the ZpQ-injective hull W of an irreducible ZpQ-module (viewed as a ZQ-module) or is the submodule W[pn] of such a W consisting of the elements ω ∈ W which satisfy pnw = 0.This classification is used to classify certain abelian-by-nilpotent groups which satisfy Min-n, the minimal condition on normal subgroups. Among the groups to which our classification applies are all quasi-radicable metabelian groups with Min-n, and all metabelian groups which satisfy Min-n and have abelian Sylow p-subgroups for all p.It is also shown that if Q is any countable locally finite p'-group and V is a ZQ-module whose additive group is a p-group, then V can be embedded in a ZQ-module whose additive group is a minimal divisible group containing that of V. Some applications of this result are given.

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.

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