THE BINARY ADDING MACHINE AND SOLVABLE GROUPS

2003 ◽  
Vol 13 (01) ◽  
pp. 95-110
Author(s):  
SAID SIDKI
Keyword(s):  
Binary Tree ◽  
Metabelian Group ◽  
Solvable Groups ◽  
Solvable Subgroup ◽  
Free Metabelian Group ◽  
Torsion Free

We prove that any solvable subgroup K of automorphisms of the binary tree, which contains the binary adding machine is an extension of a torsion-free metabelian group by a finite 2-group. If the group K is moreover nilpotent then it is torsion-free abelian.

ON TORSION-FREE METABELIAN GROUPS WITH COMMUTATOR QUOTIENTS OF PRIME EXPONENT

1999 ◽  
Vol 09 (05) ◽  
pp. 493-520 ◽  
Author(s):  
NARAIN GUPTA ◽  
SAID SIDKI
Keyword(s):  
Metabelian Group ◽  
Free Groups ◽  
Metabelian Groups ◽  
Free Metabelian Group ◽  
Torsion Free ◽  
Prime Exponent

Let G be a torsion-free metabelian group having for commutator quotient, an elementary abelian p-group of rank k. It is shown that k≥3 for all primes p. Examples of such metabelian torsion-free groups are constructed for all primes p and all ranks k≥3, except for p=2, k=3.

scholarly journals GEOMETRICAL APPROACH TO THE FREE SOLVABLE GROUPS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 1243-1260 ◽  
Author(s):  
A. M. VERSHIK ◽  
S. V. DOBRYNIN
Keyword(s):  
Cayley Graph ◽  
Cohomology Group ◽  
Normal Forms ◽  
Metabelian Group ◽  
Solvable Groups ◽  
Geometrical Approach ◽  
Metabelian Groups ◽  
Free Metabelian Group ◽  
Dimensional Lattice ◽  
Points Of View

We give a topological interpretation of the free metabelian group, following the plan described in [11, 12]. Namely, we represent the free metabelian group with d-generators as an extension of the group of the first homology of the d-dimensional lattice (as Cayley graph of the group ℤd), with a canonical 2-cocycle. This construction opens a possibility to study metabelian groups from new points of view; in particular to give useful normal forms of the elements of the group, leading to applications to the random walks, and so on. We also describe the satellite groups which correspond to all 2-cocycles of cohomology group associated with the free solvable groups. The homology of the Cayley graph can be used for studying the wide class of groups which include the class of all solvable groups.

scholarly journals A just-nonsolvable torsion-free group defined on the binary tree

1999 ◽  
Vol 211 (1) ◽  
pp. 99-114 ◽  
Author(s):  
A.M. Brunner ◽  
Said Sidki ◽  
Ana Cristina Vieira
Keyword(s):  
Free Group ◽  
Binary Tree ◽  
Torsion Free Group ◽  
Torsion Free

Some Applications of Artamonov-Quillen-Suslin Theorems to Metabelian Inner Rank and Primitivity

1994 ◽  
Vol 46 (2) ◽  
pp. 298-307 ◽  
Author(s):  
C. K. Gupta ◽  
N. D. Gupta ◽  
G. A. Noskov
Keyword(s):  
Metabelian Group ◽  
Sufficient Conditions ◽  
Surface Group ◽  
Maximal Rank ◽  
Orientable Surface ◽  
Surface Groups ◽  
Modular Element ◽  
Free Metabelian Group ◽  
One Relator Groups ◽  
Necessary And Sufficient

AbstractFor any variety of groups, the relative inner rank of a given groupG is defined to be the maximal rank of the -free homomorphic images of G. In this paper we explore metabelian inner ranks of certain one-relator groups. Using the well-known Quillen-Suslin Theorem, in conjunction with an elegant result of Artamonov, we prove that if r is any "Δ-modular" element of the free metabelian group Mn of rank n > 2 then the metabelian inner rank of the quotient group Mn/(r) is at most [n/2]. As a corollary we deduce that the metabelian inner rank of the (orientable) surface group of genus k is precisely k. This extends the corresponding result of Zieschang about the absolute inner ranks of these surface groups. In continuation of some further applications of the Quillen-Suslin Theorem we give necessary and sufficient conditions for a system g = (g1,..., gk) of k elements of a free metabelian group Mn, k ≤ n, to be a part of a basis of Mn. This extends results of Bachmuth and Timoshenko who considered the cases k = n and k < n — 3 respectively.

Wreath Product of O*-Groups that is not in O*

1971 ◽  
Vol 14 (3) ◽  
pp. 453-454
Author(s):  
S. V. Modak
Keyword(s):  
Wreath Product ◽  
Cyclic Group ◽  
Metabelian Group ◽  
Ordered Group ◽  
Infinite Cyclic Group ◽  
Free Metabelian Group ◽  
Ordered Groups

It is well known that the wreath product of two ordered groups is an ordered group. In [2] Fuchs asks if the same is true for O*-groups. Here we construct an example to show that the wreath product of an infinite cyclic group with a free metabelian group is not an O*-group.

On Torsion-Free Groups of Finite Rank

1984 ◽  
Vol 36 (6) ◽  
pp. 1067-1080 ◽  
Author(s):  
David Meier ◽  
Akbar Rhemtulla
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Finite Rank ◽  
Maximal Element ◽  
Free Groups ◽  
Solvable Groups ◽  
Finitely Generated ◽  
Isolated Subgroup ◽  
Image Position ◽  
Torsion Free

This paper deals with two conditions which, when stated, appear similar, but when applied to finitely generated solvable groups have very different effect. We first establish the notation before stating these conditions and their implications. If H is a subgroup of a group G, let denote the setWe say G has the isolator property if is a subgroup for all H ≦ G. Groups possessing the isolator property were discussed in [2]. If we define the relation ∼ on the set of subgroups of a given group G by the rule H ∼ K if and only if , then ∼ is an equivalence relation and every equivalence class has a maximal element which may not be unique. If , we call H an isolated subgroup of G.

Sign in / Sign up

Export Citation Format

Share Document