scholarly journals On an infinite integral linear group

1971 ◽  
Vol 4 (3) ◽  
pp. 321-342 ◽  
Author(s):  
I.H. Farouqi ◽  
Hanna Neumann
Keyword(s):  
Normal Subgroup ◽  
Order Type ◽  
Normal Subgroups ◽  
Congruence Subgroups ◽  
Infinite Rank ◽  
Finite Dimensional ◽  
Infinite Integral ◽  
Countably Infinite ◽  
Dimensional Integral ◽  
The Continuum

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.

scholarly journals Maximal normal subgroups of the integral linear group of countable degree

1976 ◽  
Vol 15 (3) ◽  
pp. 439-451 ◽  
Author(s):  
R.G. Burns ◽  
I.H. Farouqi
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Normal Subgroup ◽  
Vector Space ◽  
Free Abelian Group ◽  
Normal Structure ◽  
Normal Subgroups ◽  
Finite Degree ◽  
Infinite Rank ◽  
Countably Infinite

This paper continues the second author's investigation of the normal structure of the automorphism group г of a free abelian group of countably infinite rank. It is shown firstly that, in contrast with the case of finite degree, for each prime p every linear transformation of the vector space of countably infinite dimension over Zp, the field of p elements, is induced by an element of г Since by a result of Alex Rosenberg GL(אo, Zp ) has a (unique) maximal normal subgroup, it then follows that г has maximal normal subgroups, one for each prime.

Normality in Elementary Subgroups of Chevalley Groups Over Rings

1976 ◽  
Vol 28 (2) ◽  
pp. 420-428 ◽  
Author(s):  
James F. Hurley
Keyword(s):  
Normal Subgroup ◽  
Commutative Ring ◽  
Lie Algebra ◽  
Chevalley Group ◽  
Complex Field ◽  
Normal Subgroups ◽  
Chevalley Groups ◽  
Simple Lie Algebra ◽  
Elementary Subgroup ◽  
Finite Dimensional

In [6] we have constructed certain normal subgroups G7 of the elementary subgroup GR of the Chevalley group G(L, R) over R corresponding to a finite dimensional simple Lie algebra L over the complex field, where R is a commutative ring with identity. The method employed was to augment somewhat the generators of the elementary subgroup EI of G corresponding to an ideal I of the underlying Chevalley algebra LR;EI is thus the group generated by all xr(t) in G having the property that ter ⊂ I. In [6, § 5] we noted that in general EI actually had to be enlarged for a normal subgroup of GR to be obtained.

On ϕ-ideals and the structure of Lie algebras

2020 ◽  
pp. 2150193
Author(s):  
Leila Goudarzi ◽  
Zahra Riyahi
Keyword(s):  
Normal Subgroup ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Normal Subgroups ◽  
Basic Properties ◽  
Finite Dimensional

This paper aims to study the concept of [Formula: see text]-ideals of a finite-dimensional Lie algebra which is analogous to the concept of [Formula: see text]-ideal and [Formula: see text]-normal subgroup. We compile some basic properties of [Formula: see text]-ideals and consider the influence of this concept on the structure of a finite-dimensional Lie algebra, especially its solvability and supersolvability. We also show that the counterpart of some results for [Formula: see text]-ideals of Lie algebras and [Formula: see text]-normal subgroups are not valid here.

scholarly journals Normal subgroups in the group of column-finite infinite matrices

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Waldemar Hołubowski ◽  
Martyna Maciaszczyk ◽  
Sebastian Zurek
Keyword(s):  
Normal Subgroup ◽  
Vector Space ◽  
Division Ring ◽  
Classical Result ◽  
Normal Subgroups ◽  
Linear Transformations ◽  
Finite Dimensional ◽  
Infinite Cardinality ◽  
Positive Integers ◽  
Dimensional Range

Abstract The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of GL ⁢ ( n , K ) \mathrm{GL}(n,K) , where 𝐾 is a field and n ≥ 3 n\geq 3 , which is not contained in the center contains SL ⁢ ( n , K ) \mathrm{SL}(n,K) . Rosenberg described the normal subgroups of GL ⁢ ( V ) \mathrm{GL}(V) , where 𝑉 is a vector space of any infinite cardinality dimension over a division ring. However, when he considers subgroups of the direct product of the center and the group of linear transformations 𝑔 such that g - id V g-\mathrm{id}_{V} has finite-dimensional range, the proof is incomplete. We fill this gap for countably dimensional 𝑉 giving description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any field.

scholarly journals The normal subgroup structure of the infinite general linear group

1981 ◽  
Vol 24 (3) ◽  
pp. 197-202 ◽  
Author(s):  
David G. Arrell
Keyword(s):  
Normal Subgroup ◽  
Linear Group ◽  
General Linear Group ◽  
Complete Classification ◽  
General Linear ◽  
Normal Subgroups ◽  
Finiteness Conditions ◽  
Finite Dimensional ◽  
Finite Dimensional Case

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).

scholarly journals Normal subgroups of diffeomorphism and homeomorphism groups of ℝn and other open manifolds

2011 ◽  
Vol 31 (6) ◽  
pp. 1835-1847 ◽  
Author(s):  
PAUL A. SCHWEITZER, S. J.
Keyword(s):  
Normal Subgroup ◽  
Compact Manifold ◽  
Normal Subgroups ◽  
Open Manifold ◽  
Open Manifolds ◽  
Group Of Homeomorphisms ◽  
Homeomorphism Groups

AbstractWe determine all the normal subgroups of the group of Cr diffeomorphisms of ℝn, 1≤r≤∞, except when r=n+1 or n=4, and also of the group of homeomorphisms of ℝn ( r=0). We also study the group A0 of diffeomorphisms of an open manifold M that are isotopic to the identity. If M is the interior of a compact manifold with non-empty boundary, then the quotient of A0 by the normal subgroup of diffeomorphisms that coincide with the identity near to a given end e of M is simple.

scholarly journals On groups, whose non-normal subgroups are either contranormal or core-free

2020 ◽  
pp. 3-8
Author(s):  
L.A. Kurdachenko ◽  
◽  
A.A. Pypka ◽  
I.Ya. Subbotin ◽  
◽  
...  
Keyword(s):  
Normal Subgroup ◽  
Normal Subgroups

We investigate the influence of some natural types of subgroups on the structure of groups. A subgroup H of a group G is called contranormal in G, if G = HG. A subgroup H of a group G is called core-free in G, if CoreG(H) =〈1〉. We study the groups, in which every non-normal subgroup is either contranormal or core-free. In particular, we obtain the structure of some monolithic and non-monolithic groups with this property

scholarly journals Generating infinite monoids of cellular automata

2021 ◽  
pp. 2250215
Author(s):  
Alonso Castillo-Ramirez
Keyword(s):  
Cellular Automata ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Group Of Units ◽  
Finite Dimensional ◽  
Index Condition ◽  
Indicable Group

For a group [Formula: see text] and a set [Formula: see text], let [Formula: see text] be the monoid of all cellular automata over [Formula: see text], and let [Formula: see text] be its group of units. By establishing a characterization of surjunctive groups in terms of the monoid [Formula: see text], we prove that the rank of [Formula: see text] (i.e. the smallest cardinality of a generating set) is equal to the rank of [Formula: see text] plus the relative rank of [Formula: see text] in [Formula: see text], and that the latter is infinite when [Formula: see text] has an infinite decreasing chain of normal subgroups of finite index, condition which is satisfied, for example, for any infinite residually finite group. Moreover, when [Formula: see text] is a vector space over a field [Formula: see text], we study the monoid [Formula: see text] of all linear cellular automata over [Formula: see text] and its group of units [Formula: see text]. We show that if [Formula: see text] is an indicable group and [Formula: see text] is finite-dimensional, then [Formula: see text] is not finitely generated; however, for any finitely generated indicable group [Formula: see text], the group [Formula: see text] is finitely generated if and only if [Formula: see text] is finite.

On normal subgroups of M-groups

2019 ◽  
Vol 18 (04) ◽  
pp. 1950074
Author(s):  
Xuewu Chang
Keyword(s):  
Normal Subgroup ◽  
Solvable Group ◽  
Solvable Groups ◽  
Hall Subgroup ◽  
Embedding Theorem ◽  
The Other ◽  
Finite Solvable Group ◽  
Embedding Problem ◽  
Normal Subgroups ◽  
Other Hand

The normal embedding problem of finite solvable groups into [Formula: see text]-groups was studied. It was proved that for a finite solvable group [Formula: see text], if [Formula: see text] has a special normal nilpotent Hall subgroup, then [Formula: see text] cannot be a normal subgroup of any [Formula: see text]-group; on the other hand, if [Formula: see text] has a maximal normal subgroup which is an [Formula: see text]-group, then [Formula: see text] can occur as a normal subgroup of an [Formula: see text]-group under some suitable conditions. The results generalize the normal embedding theorem on solvable minimal non-[Formula: see text]-groups to arbitrary [Formula: see text]-groups due to van der Waall, and also cover the famous counterexample given by Dade and van der Waall independently to the Dornhoff’s conjecture which states that normal subgroups of arbitrary [Formula: see text]-groups must be [Formula: see text]-groups.

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