Normal Subgroup Structure Research Articles

Anisotropic groups of type Anand the commuting graph of finite simple groups

In this paper we make a contribution to the MargulisPlatonov conjecture, which describes the normal subgroup structure of algebraic groups over number fields. We establish the conjecture for inner forms of anisotropic groups of type An. We obtain information on the commuting graph of nonabelian finite simple groups, and consequently, using the paper by Segev, 1999, we obtain results on the normal structure and quotient groups of the multiplicative group of a division algebra.

Normal Subgroups in Substitution Groups of Formal Power Series

Let R be a commutative ring with 1, and let N(R)≔t+t2R[t] be the group of normalized formal power series over R under substitution. In this paper we investigate the connection between the ideal structure of R and the normal subgroup structure of N(R). In particular, we show that, if K is a finite field of characteristic not equal to two, then every proper quotient group of the so-called Nottingham group N(K) is finite. As a further application we consider the profinite completion of the group N(R). We show that, if every additive subgroup of finite index in R contains an ideal of finite index in R, then N(R̂)≅N(R̂).

The finite quotients of the multiplicative group of a division algebra of degree 3 are solvable

LetD be a finite dimensional division algebra. It is known that in a variety of cases, questions about the normal subgroup structure ofDx (the multiplicative group ofD) can be reduced to questions about finite quotients ofDx. In this paper we prove that when deg(D)=3, finite quotients ofDx are solvable. the proof uses Wedderburn’s Factorization Theorem.

Normal subgroups ofSL 1,D and the classification of finite simple groups

LetD be a division algebra of degree three over an algebraic number fieldK and let G = SLD. We prove that the normal subgroup structure of G(K) is given by the Platonov-Margulis conjecture. The proof uses the classification of finite simple groups.

On unitary groups modulo infinitesimals

We deal with a hermitian space V over an involutorial division ring D with characteristic not 2 and with a nontrivial nonarchimedean valuation. We are concerned with the group C of unitary transformations modulo the subgroup 1 + J ( J is the ideal of nonunits of the valuation ring in D). The normal subgroup structure of C is studied. It is also shown that the group C of a hyperbolic space is generated by certain standard matrices.

ON THE REDUCED NORM 1 GROUP OF A DIVISION ALGEBRA OVER A GLOBAL FIELD

It is proved that if the Platonov-Margulis conjecture on the standard structure of normal subgroups holds for the division algebras of index r, then it also holds for the division algebras of index n=2mr, for any m. Thus the conjecture is proved for the division algebras of index n=2m, for any m, and its proof in the general case is reduced to the case of division algebras of odd index.

Subgroups of semidirect products

The notion of a reduced crossed homomorphism is introduced and subgroups of a semidirect product are described by means of it. Subsemidirect products and semidirect products with a given structure of normal subgroups are characterized.

The Normal Subgroup Structure of the Picard Group

The Normal Subgroup Structure of the Picard Group

The normal subgroup structure of the Picard group

The Picard group Γ \Gamma is P S L 2 ( Z [ i ] ) PS{L_2}(Z[i]) , the group of linear fractional transformations with Gaussian integer coefficients. We examine the structure of the normal subgroups of Γ \Gamma . In particular we give a complete classification of the normal subgroups for indices less than 60 60 and show that beyond this there are large gaps in the possible indices. This classification depends on the structure of the derived series. Finally we give examples of normal noncongruence subgroups.

The normal subgroup structure of the infinite general linear group

The classification of the normal subgroups of the infinite general linear group GL(Ω, R) has received much attention and has been studied in, for example, (6), (4) and (2). The main theorem of (6) gives a complete classification of the normal subgroups of GL(Ω, R) when R is a division ring, while the results of (2) require that R satisfies certain finiteness conditions. The object of this paper is to produce a classification, along the lines of that given by Wilson in (7) or by Bass in (3) in the finite dimensional case, that does not require any finiteness assumptions. However, when R is Noetherian, the classification given here reduces to that given in (2).

On the structure of normal subgroups of monomial groups

On the structure of normal subgroups of monomial groups

On an infinite integral linear group

This paper investigates the normal subgroup structure of the automorphism group Γ of a free abelian group A of countably infinite rank. The finitary automorphisms, that is those acting non-trivially only on a direct summand of A of finite rank, form a normal subgroup Φ of Γ; the sublattice of all normal subgroups of Γ contained in Φ is in fact the sublattice of normal subgroups of Φ and has a quite transparent structure. By contrast there is a profusion of normal subgroups of Γ not contained in Φ. For example, the collection of certain types of these normal subgroups, defined as generalizations of the congruence subgroups of finite dimensional integral linear groups, if partially ordered by inclusion, can be shown to contain infinitely many chains of the order type of the continuum.