TREE-WREATHING APPLIED TO GENERATION OF GROUPS BY FINITE AUTOMATA

2005 ◽  
Vol 15 (05n06) ◽  
pp. 1205-1212 ◽  
Author(s):  
SAID SIDKI
Keyword(s):  
Wreath Product ◽  
Finite Index ◽  
Binary Tree ◽  
Infinite Sequence ◽  
Finite Automata ◽  
Residually Finite ◽  
Group Of Automorphisms ◽  
Triangular Group ◽  
Product Of Groups

We extend the tree-wreath product of groups introduced by Brunner and the author, as a generalization of the restricted wreath product, thus enlarging the class of groups known to be generated by finite synchronous automata. In particular, we prove that given a countable abelian residually finite 2 -group H and B = B(n,ℤ), a canonical subgroup of finite index in GL(n,ℤ), then the restricted wreath product H wr B can be generated by finite synchronous automata on 0,1. This is obtained by producing a representation of B as a group of automorphisms of the binary tree such that the stabilizer of the infinite sequence of 0's is trivial. The uni-triangular group U = U(n,ℤ) is a subgroup of B(n,ℤ) and so, H wr U also can be generated by finite synchronous automata on 0,1.

scholarly journals Residually finite dimensional algebras and polynomial almost identities

2020 ◽  
pp. 2250038
Author(s):  
Michael Larsen ◽  
Aner Shalev
Keyword(s):  
Lie Algebra ◽  
Finite Index ◽  
Open Group ◽  
Finite Codimension ◽  
Finite Dimensional Algebra ◽  
Nilpotent Ideal ◽  
Dimensional Algebra ◽  
Residually Finite ◽  
Finite Dimensional ◽  
Theoretic Problem

Let [Formula: see text] be a residually finite dimensional algebra (not necessarily associative) over a field [Formula: see text]. Suppose first that [Formula: see text] is algebraically closed. We show that if [Formula: see text] satisfies a homogeneous almost identity [Formula: see text], then [Formula: see text] has an ideal of finite codimension satisfying the identity [Formula: see text]. Using well known results of Zelmanov, we conclude that, if a residually finite dimensional Lie algebra [Formula: see text] over [Formula: see text] is almost [Formula: see text]-Engel, then [Formula: see text] has a nilpotent (respectively, locally nilpotent) ideal of finite codimension if char [Formula: see text] (respectively, char [Formula: see text]). Next, suppose that [Formula: see text] is finite (so [Formula: see text] is residually finite). We prove that, if [Formula: see text] satisfies a homogeneous probabilistic identity [Formula: see text], then [Formula: see text] is a coset identity of [Formula: see text]. Moreover, if [Formula: see text] is multilinear, then [Formula: see text] is an identity of some finite index ideal of [Formula: see text]. Along the way we show that if [Formula: see text] has degree [Formula: see text], and [Formula: see text] is a finite [Formula: see text]-algebra such that the probability that [Formula: see text] (where [Formula: see text] are randomly chosen) is at least [Formula: see text], then [Formula: see text] is an identity of [Formula: see text]. This solves a ring-theoretic analogue of a (still open) group-theoretic problem posed by Dixon,

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.

scholarly journals The multiplicator of a regular product of groups

1972 ◽  
Vol 7 (2) ◽  
pp. 279-296 ◽  
Author(s):  
William Haebich ◽  
P.J. Cossey
Keyword(s):  
Wreath Product ◽  
Product Of Groups

It is shown that if G is an arbitrary regular product of its subgroups Aλ, ∈ ϵ I, then the multiplicator, M(G), is the director product of the M(Aλ) together with a certain other group. This extends a calculation of M(A1 × A2) due to Schur. As an application, we find the multiplicator of a vertai wreath product A wrVB where A is abelian. A representing group for a finite regular product is also constructed.

A Conjecture of Bachmuth and Mochizuki on Automorphisms of Soluble Groups

1976 ◽  
Vol 28 (6) ◽  
pp. 1302-1310 ◽  
Author(s):  
Brian Hartley
Keyword(s):  
Finite Index ◽  
Soluble Group ◽  
Abelian Groups ◽  
Free Subgroup ◽  
Linear Groups ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Group Of Automorphisms ◽  
Finitely Generated Group ◽  
Soluble Subgroup

In [1], Bachmuth and Mochizuki conjecture, by analogy with a celebrated result of Tits on linear groups [8], that a finitely generated group of automorphisms of a finitely generated soluble group either contains a soluble subgroup of finite index (which may of course be taken to be normal) or contains a non-abelian free subgroup. They point out that their conjecture holds for nilpotent-by-abelian groups and in some other cases.

The Symplectic Representation and the Torelli Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Finite Index ◽  
Algebraic Structure ◽  
Symplectic Form ◽  
Intersection Number ◽  
Free Subgroup ◽  
Torelli Group ◽  
Residually Finite ◽  
Basic Properties ◽  
Torsion Free Subgroup ◽  
Torsion Free

This chapter discusses the basic properties and applications of a symplectic representation, denoted by Ψ‎, and its kernel, called the Torelli group. After describing the algebraic intersection number as a symplectic form, the chapter presents three different proofs of the surjectivity of Ψ‎, each illustrating a different theme. It also illustrates the usefulness of the symplectic representation by two applications to understanding the algebraic structure of Mod(S). First, the chapter explains how this representation is used by Serre to prove the theorem that Mod(Sɡ) has a torsion-free subgroup of finite index. It thens uses the symplectic representation to prove, following Ivanov, the following theorem of Grossman: Mod(Sɡ) is residually finite. It also considers some of the pioneering work of Dennis Johnson on the Torelli group. In particular, a Johnson homomorphism is constructed and some of its applications are given.

Wreath products by a leavitt path algebra and affinizations

2014 ◽  
Vol 24 (05) ◽  
pp. 707-714 ◽  
Author(s):  
Adel Alahmadi ◽  
Hamed Alsulami
Keyword(s):  
New York ◽  
Wreath Product ◽  
Wreath Products ◽  
Path Algebra ◽  
Leavitt Path Algebra ◽  
Quotient Graph ◽  
Affine Algebras ◽  
Property B ◽  
Associate Algebra ◽  
Product Of Groups

We introduce ring theoretic constructions that are similar to the construction of wreath product of groups [M. Kargapolov and Y. Merzlyakov, Fundamentals of the Theory of Groups (Springer-Verlag, New York, 1979)]. In particular, for a given graph Γ = (V, E) and an associate algebra A, we construct an algebra B = A wr L(Γ) with the following property: B has an ideal I, which consists of (possibly infinite) matrices over A, B/I ≅ L(Γ), the Leavitt path algebra of the graph Γ. Let W ⊂ V be a hereditary saturated subset of the set of vertices [G. Abrams and G. Aranda Pino, The Leavitt path algebra of a graph, J. Algebra 293(2) (2005) 319–334], Γ(W) = (W, E(W, W)) is the restriction of the graph Γ to W, Γ/W is the quotient graph [G. Abrams and G. Aranda Pino, The Leavitt path algebra of a graph, J. Algebra 293(2) (2005) 319–334]. Then L(Γ) ≅ L(W) wr L(Γ/W). As an application we use wreath products to construct new examples of (i) affine algebras with non-nil Jacobson radicals, (ii) affine algebras with non-nilpotent locally nilpotent radicals.

scholarly journals On Finitely Generated Subgroups of Free Products

1971 ◽  
Vol 12 (3) ◽  
pp. 358-364 ◽  
Author(s):  
R. G. Burns
Keyword(s):  
Finite Index ◽  
Finitely Generated ◽  
Free Products ◽  
Residually Finite ◽  
Usual Definition

IfHis a subgroup of a groupGwe shall say thatGisH-residually finiteif for every elementginG, outsideH, there is a subgroup of finite index inG, containingHand still avoidingg. (Then, according to the usual definition,Gisresidually finiteif it isE-residually finite, whereEis the identity subgroup). Definitions of other terms used below may be found in § 2 or in [6].

Groups Whose Subgroup Growth is Less than Linear

1997 ◽  
Vol 07 (01) ◽  
pp. 77-91 ◽  
Author(s):  
Aner Shalev
Keyword(s):  
Finite Group ◽  
Finite Index ◽  
Residually Finite ◽  
Subgroup Growth ◽  
Residually Finite Group ◽  
Finite Quotients

Let G be a residually finite group and let an(G) denote the number of index n subgroups of G. It is shown that an(G)/n →0 if and only if G has a finite index central subgroup whose finite quotients are all cyclic. As an application we show that the degree of a group of polynomial subgroup growth cannot lie strictly between 0 and 1.

scholarly journals Near Frattini subgroups of residually finite generalized free products of groups

2001 ◽  
Vol 26 (2) ◽  
pp. 117-121
Author(s):  
Mohammad K. Azarian
Keyword(s):  
Free Product ◽  
Finite Index ◽  
Frattini Subgroup ◽  
Free Products ◽  
Residually Finite ◽  
Identical Relation ◽  
Products Of Groups ◽  
Generalized Free Product ◽  
Amalgamated Subgroup ◽  
Generalized Free Products

LetG=A★HBbe the generalized free product of the groupsAandBwith the amalgamated subgroupH. Also, letλ(G)andψ(G)represent the lower near Frattini subgroup and the near Frattini subgroup ofG, respectively. IfGis finitely generated and residually finite, then we show thatψ(G)≤H, providedHsatisfies a nontrivial identical relation. Also, we prove that ifGis residually finite, thenλ(G)≤H, provided: (i)Hsatisfies a nontrivial identical relation andA,Bpossess proper subgroupsA1,B1of finite index containingH; (ii) neitherAnorBlies in the variety generated byH; (iii)H<A1≤AandH<B1≤B, whereA1andB1each satisfies a nontrivial identical relation; (iv)His nilpotent.

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