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.

scholarly journals APPROXIMATING THE FIRST L2-BETTI NUMBER OF RESIDUALLY FINITE GROUPS

2011 ◽  
Vol 03 (02) ◽  
pp. 153-160 ◽  
Author(s):  
W. LÜCK ◽  
D. OSIN
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Betti Number ◽  
Betti Numbers ◽  
Torsion Group ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Residually Finite Groups

We show that the first L2-betti number of a finitely generated residually finite group can be estimated from below by using ordinary first betti numbers of finite index normal subgroups. As an application, we construct a finitely generated infinite residually finite torsion group with positive first L2-betti number.

Spectral synthesis and residually finite-dimensional algebras

2017 ◽  
Vol 16 (10) ◽  
pp. 1750200 ◽  
Author(s):  
László Székelyhidi ◽  
Bettina Wilkens
Keyword(s):  
Spectral Analysis ◽  
Abelian Group ◽  
Theoretical Approach ◽  
Abelian Groups ◽  
Spectral Synthesis ◽  
Residually Finite ◽  
Finite Dimensional ◽  
Analysis And Synthesis ◽  
Spectral Analysis And Synthesis

In 2004, a counterexample was given for a 1965 result of R. J. Elliott claiming that discrete spectral synthesis holds on every Abelian group. Since then the investigation of discrete spectral analysis and synthesis has gained traction. Characterizations of the Abelian groups that possess spectral analysis and spectral synthesis, respectively, were published in 2005. A characterization of the varieties on discrete Abelian groups enjoying spectral synthesis is still missing. We present a ring theoretical approach to the issue. In particular, we provide a generalization of the Principal Ideal Theorem on discrete Abelian groups.

scholarly journals Locally graded groups with all subgroups normal-by-finite

1996 ◽  
Vol 60 (2) ◽  
pp. 222-227 ◽  
Author(s):  
Howard Smith ◽  
James Wiegold
Keyword(s):  
Finite Group ◽  
Finite Index ◽  
Finitely Generated ◽  
Locally Finite ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Open Question ◽  
Locally Graded Groups

AbstractIn a paper published in this journal [1], J. T. Buckely, J. C. Lennox, B. H. Neumann and the authors considered the class of CF-groups, that G such that |H: CoreG (H)| is finite for all subgroups H. It is shown that locally finite CF-groups are abelian-by-finite and BCF, that is, there is an integer n such that |H: CoreG(H)| ≤ n for all subgroups H. The present paper studies these properties in the class of locally graded groups, the main result being that locally graded BCF-groups are abelian-by-finite. Whether locally graded CF-groups are BFC remains an open question. In this direction, the following problems is posed. Does there exist a finitely generated infinite periodic residually finite group in which all subgroups are finite or of finite index? Such groups are locally graded and CF but not BCF.

NILPOTENT LINEAR SEMIGROUPS

2006 ◽  
Vol 16 (01) ◽  
pp. 141-160 ◽  
Author(s):  
ERIC JESPERS ◽  
DAVID RILEY
Keyword(s):  
Finite Group ◽  
Lie Algebra ◽  
Associative Algebra ◽  
Burnside Problem ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Adjoint Semigroup ◽  
Restricted Burnside Problem ◽  
Global And Local

We characterize the structure of linear semigroups satisfying certain global and local nilpotence conditions and deduce various Engel-type results. For example, using a form of Zel'manov's solution of the restricted Burnside problem we are able to show that a finitely generated residually finite group is nilpotent if and only if it satisfies a certain 4-generator property of semigroups we call WMN. Methods of linear semigroups then allow us to prove that a linear semigroup is Mal'cev nilpotent precisely when it satisfies WMN. As an application, we show that a finitely generated associative algebra is nilpotent when viewed as a Lie algebra if and only if its adjoint semigroup is WMN.

scholarly journals Some notes on Lie ideals in division rings

2018 ◽  
Vol 17 (03) ◽  
pp. 1850049
Author(s):  
M. Aaghabali ◽  
M. Ariannejad ◽  
A. Madadi
Keyword(s):  
Division Ring ◽  
Commutator Subgroup ◽  
Product Formula ◽  
Main Concern ◽  
Normal Subgroups ◽  
Lie Ideal ◽  
Additive Subgroup ◽  
Finitely Generated ◽  
Finite Dimensional ◽  
Lie Product Formula

A Lie ideal of a division ring [Formula: see text] is an additive subgroup [Formula: see text] of [Formula: see text] such that the Lie product [Formula: see text] of any two elements [Formula: see text] is in [Formula: see text] or [Formula: see text]. The main concern of this paper is to present some properties of Lie ideals of [Formula: see text] which may be interpreted as being dual to known properties of normal subgroups of [Formula: see text]. In particular, we prove that if [Formula: see text] is a finite-dimensional division algebra with center [Formula: see text] and [Formula: see text], then any finitely generated [Formula: see text]-module Lie ideal of [Formula: see text] is central. We also show that the additive commutator subgroup [Formula: see text] of [Formula: see text] is not a finitely generated [Formula: see text]-module. Some other results about maximal additive subgroups of [Formula: see text] and [Formula: see text] are also presented.

scholarly journals Some properties of endomorphisms in residually finite groups

1977 ◽  
Vol 24 (1) ◽  
pp. 117-120 ◽  
Author(s):  
Ronald Hirshon
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Free Product ◽  
Finite Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Residually Finite Groups ◽  
A Finite Group ◽  
Torsion Free

AbstractIf ε is an endomorphism of a finitely generated residually finite group onto a subgroup Fε of finite index in F, then there exists a positive integer k such that ε is an isomorphism of Fεk. If K is the kernel of ε, then K is a finite group so that if F is a non trivial free product or if F is torsion free, then ε is an isomorphism on F. If ε is an endomorphism of a finitely generated resedually finite group onto a subgroup Fε (not necessatily of ginite index in F) and if the kernel of ε obeys the minimal condition for subgroups then there exists a positive integer k such that ε is an isomorphism on Fεk.

scholarly journals Projectivity of Hopf algebras over subalgebras with semilocal central localizations

2008 ◽  
Vol 2 (1) ◽  
pp. 1-40 ◽  
Author(s):  
Serge Skryabin
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Module Algebra ◽  
Finitely Generated ◽  
Residually Finite ◽  
Hopf Subalgebra ◽  
Finite Dimensional ◽  
Algebra Over A Field

AbstractThe purpose of this paper is to extend the class of pairs A, H where H is a Hopf algebra over a field and A a right coideal subalgebra for which H is proved to be either projective or flat as an A-module. The projectivity is obtained under the assumption that H is residually finite dimensional, A has semilocal localizations with respect to a central subring, and there exists a Hopf subalgebra B of H such that the antipode of B is bijective and B is a finitely generated A-module. The flatness of H over A is shown to hold when H is a directed union of residually finite dimensional Hopf subalgebras, and there exists a Hopf subalgebra of H whose center contains A. More general projectivity and flatness results are established for (co)equivariant modules over an H-(co)module algebra under similar assumptions.

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

1982 ◽  
Vol 25 (1) ◽  
pp. 81-86 ◽  
Author(s):  
David G. Arrell
Keyword(s):  
General Linear Group ◽  
Subnormal Subgroup ◽  
Complete Classification ◽  
Normal Subgroups ◽  
Finiteness Conditions ◽  
Group Of Units ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Infinite Set

Let R be a ring with identity, let Ω be an infinite set and let M be the free R-module R(Ω). In [1] we investigated the problem of locating and classifying the normal subgroups of GL(Ω, R), the group of units of the endomorphism ring EndRM, where R was an arbitrary ring with identity. (This extended the work of [3] and [8] where it was necessary for R to satisfy certain finiteness conditions.) When R is a division ring, the complete classification of the normal subgroups of GL(Ω, R) is given in [9] and the corresponding results for a Hilbert space are given in [6] and [7]. The object of this paper is to extend the methods of [1] to yield a classification of the subnormal subgroups of GL(Ω, R) along the lines of that given by Wilson in [10] in the finite dimensional case.

scholarly journals Some topological properties of residually Černikov groups

1982 ◽  
Vol 23 (1) ◽  
pp. 65-82 ◽  
Author(s):  
M. R. Dixon
Keyword(s):  
Finite Groups ◽  
Inverse Limit ◽  
Normal Subgroups ◽  
Topological Properties ◽  
Profinite Groups ◽  
Dense Subgroup ◽  
Locally Finite ◽  
Soluble Groups ◽  
Residually Finite ◽  
Residually Finite Group

In this paper we shall indicate how to generalise the concept of a cofinite group (see [7]). We recall that any residually finite group can be made into a topological group by taking as a basis of neighbourhoods of the identity precisely the normal subgroups of finite index. The class of compact cofinite groups is then easily seen to be the class of profinite groups, where a group is profinite if and only if it is an inverse limit of finite groups. It turns out that every cofinite group can be embedded as a dense subgroup of a profinite group. This has important consequences for the class of countable locally finite-soluble groups with finite Sylow p-subgroups for all primes p, as shown in [7] and [14].

On R0-Closed Classes, and Finitely Generated Groups

1970 ◽  
Vol 22 (1) ◽  
pp. 176-184 ◽  
Author(s):  
Rex Dark ◽  
Akbar H. Rhemtulla
Keyword(s):  
Normal Subgroups ◽  
Finitely Generated ◽  
Maximal Condition ◽  
Soluble Groups ◽  
Finitely Generated Group ◽  
Finitely Generated Groups ◽  
Central Factors ◽  
Subnormal Subgroups

1.1. If a group satisfies the maximal condition for normal subgroups, then all its central factors are necessarily finitely generated. In [2], Hall asked whether there exist finitely generated soluble groups which do not satisfy the maximal condition for normal subgroups but all of whose central factors are finitely generated. We shall answer this question in the affirmative. We shall also construct a finitely generated group all of whose subnormal subgroups are perfect (and which therefore has no non-trivial central factors), but which does not satisfy the maximal condition for normal subgroups. Related to these examples is the question of which classes of finitely generated groups satisfy the maximal condition for normal subgroups. A characterization of such classes has been obtained by Hall, and we shall include his result as our first theorem.

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