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.

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.

Periodic normal subgroups of linear groups

1998 ◽  
Vol 71 (3) ◽  
pp. 169-172 ◽  
Author(s):  
B.A.F. Wehrfritz
Keyword(s):  
Linear Groups ◽  
Normal Subgroups

Unipotent normal subgroups of skew linear groups

1989 ◽  
Vol 105 (1) ◽  
pp. 37-51 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Normal Subgroup ◽  
Matrix Group ◽  
Linear Groups ◽  
Normal Subgroups ◽  
Matrix Groups ◽  
Division Rings ◽  
Additional Assumptions

A recurrent problem over many years in the study of linear groups has been the determination of the central height of a unipotent normal subgroup of some matrix group of specified type. In the theory of matrix groups over division rings, unipotent elements frequently present special difficulties and these have usually been by-passed by the addition of some suitable hypothesis. In this paper we make a start on the removal of these extraneous hypotheses. Our motivation for doing this now conies from [9], where by 3·7 of that paper the additional assumptions have finally reduced us to degree one, a situation where unipotent elements present few problems!

Group Algebras: Normal Subgroups and Ideals

2007 ◽  
Vol 75 (1) ◽  
pp. 323-332
Author(s):  
Rüdiger Göbel ◽  
Otto H. Kegel
Keyword(s):  
Group Algebras ◽  
Normal Subgroups

scholarly journals SOME INFINITE PERMUTATION GROUPS AND RELATED FINITE LINEAR GROUPS

2016 ◽  
Vol 102 (1) ◽  
pp. 136-149 ◽  
Author(s):  
PETER M. NEUMANN ◽  
CHERYL E. PRAEGER ◽  
SIMON M. SMITH
Keyword(s):  
Normal Subgroup ◽  
Vector Space ◽  
Finite Rank ◽  
Minimal Condition ◽  
Permutation Groups ◽  
Linear Groups ◽  
Normal Subgroups ◽  
Second Variation ◽  
Formula Group ◽  
Finite Linear

This article began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-n, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a $p$-adic vector space associated with $M$. This leads to our second variation, which is a study of the finite linear groups that can arise.

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