ON GROUPS ACTING ON TREES OF INFINITE CYCLIC SUBGROUPS OF FINITE INDEX

2016 ◽  
Vol 19 (1) ◽  
pp. 69-78
Author(s):  
Rasheed M. S. Mahmood ◽  
Nidal Anakira
Keyword(s):  
Finite Index ◽  
Cyclic Subgroups ◽  
Groups Acting On Trees

scholarly journals Virtual Retraction Properties in Groups

2019 ◽  
Author(s):  
Ashot Minasyan
Keyword(s):  
Finite Index ◽  
Wreath Products ◽  
Graph Products ◽  
Index Subgroup ◽  
Finitely Generated ◽  
Simple Criterion ◽  
Cyclic Subgroups ◽  
Group A ◽  
Finite Index Subgroup ◽  
Virtually Free Group

Abstract If $G$ is a group, a virtual retract of $G$ is a subgroup, which is a retract of a finite index subgroup. Most of the paper focuses on two group properties: property (LR), that all finitely generated subgroups are virtual retracts; and property (VRC), that all cyclic subgroups are virtual retracts. We study the permanence of these properties under commensurability, amalgams over retracts, graph products, and wreath products. In particular, we show that (VRC) is stable under passing to finite index overgroups, while (LR) is not. The question whether all finitely generated virtually free groups satisfy (LR) motivates the remaining part of the paper, studying virtual free factors of such groups. We give a simple criterion characterizing when a finitely generated subgroup of a virtually free group is a free factor of a finite index subgroup. We apply this criterion to settle a conjecture of Brunner and Burns.

scholarly journals Finitely generated subgroups of amalgamated free products and HNN groups

1976 ◽  
Vol 22 (3) ◽  
pp. 274-281 ◽  
Author(s):  
Daniel E. Cohen
Keyword(s):  
Free Product ◽  
Finite Index ◽  
Subnormal Subgroup ◽  
Finitely Generated ◽  
Free Products ◽  
Amalgamated Free Products ◽  
Amalgamated Free Product ◽  
Hnn Extension ◽  
Groups Acting On Trees

AbstractThe theory of groups acting on trees due to Bass and Serre (1969) is applied to simplify some results of Burns (1972, 1973) giving conditions under which an amalgamated free product or HNN extension has the properties that any finitely generated subgroup containing an infinite subnormal subgroup must have finite index and that the intersection of two finitely generated subgroups is finitely generated.

An Induction Theorem for Units of p-Adic Group Rings

1991 ◽  
Vol 34 (1) ◽  
pp. 31-35 ◽  
Author(s):  
A. K. Bhandari ◽  
S. K. Sehgal
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Index ◽  
Group Rings ◽  
Cyclic Subgroups ◽  
The Family ◽  
A Finite Group

AbstractLet G be a finite group and let C be the family of cyclic subgroups of G. We show that the normal subgroup H of U = U(ZpG) generated by U(ZpC), C ∊ C, where Zp is the ring of p-adic integers, is of finite index in U.

scholarly journals CHABAUTY LIMITS OF SIMPLE GROUPS ACTING ON TREES

2018 ◽  
Vol 19 (4) ◽  
pp. 1093-1120
Author(s):  
Pierre-Emmanuel Caprace ◽  
Nicolas Radu
Keyword(s):  
Finite Index ◽  
Automorphism Groups ◽  
Open Subgroup ◽  
Simple Groups ◽  
Finite Tree ◽  
Hausdorff Topology ◽  
Locally Finite ◽  
Isomorphism Classes ◽  
Bounded Number ◽  
Groups Acting On Trees

Let $T$ be a locally finite tree without vertices of degree $1$. We show that among the closed subgroups of $\text{Aut}(T)$ acting with a bounded number of orbits, the Chabauty-closure of the set of topologically simple groups is the set of groups without proper open subgroup of finite index. Moreover, if all vertices of $T$ have degree ${\geqslant}3$, then the set of isomorphism classes of topologically simple closed subgroups of $\text{Aut}(T)$ acting doubly transitively on $\unicode[STIX]{x2202}T$ carries a natural compact Hausdorff topology inherited from Chabauty. Some of our considerations are valid in the context of automorphism groups of locally finite connected graphs. Applications to Weyl-transitive automorphism groups of buildings are also presented.

Automorphisms of finite order of nilpotent groups II

2014 ◽  
Vol 51 (4) ◽  
pp. 547-555 ◽  
Author(s):  
B. Wehrfritz
Keyword(s):  
Nilpotent Group ◽  
Finite Order ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

scholarly journals On the sandpile model of modified wheels II

2020 ◽  
Vol 18 (1) ◽  
pp. 1531-1539
Author(s):  
Zahid Raza ◽  
Mohammed M. M. Jaradat ◽  
Mohammed S. Bataineh ◽  
Faiz Ullah
Keyword(s):  
Direct Product ◽  
Critical Group ◽  
Complete Structure ◽  
Sandpile Model ◽  
Cyclic Subgroups ◽  
Algebr Comb ◽  
Abelian Sandpile ◽  
Chip Firing ◽  
Sandpile Group

Abstract We investigate the abelian sandpile group on modified wheels {\hat{W}}_{n} by using a variant of the dollar game as described in [N. L. Biggs, Chip-Firing and the critical group of a graph, J. Algebr. Comb. 9 (1999), 25–45]. The complete structure of the sandpile group on a class of graphs is given in this paper. In particular, it is shown that the sandpile group on {\hat{W}}_{n} is a direct product of two cyclic subgroups generated by some special configurations. More precisely, the sandpile group on {\hat{W}}_{n} is the direct product of two cyclic subgroups of order {a}_{n} and 3{a}_{n} for n even and of order {a}_{n} and 2{a}_{n} for n odd, respectively.

On metalinear ETOL systems

1980 ◽  
Vol 3 (1) ◽  
pp. 15-36
Author(s):  
Grzegorz Rozenberg ◽  
Dirk Vermeir
Keyword(s):  
Finite Index ◽  
Closure Properties ◽  
Pumping Lemma ◽  
L Systems ◽  
Structural Characterizations

The concept of metalinearity in ETOL systems is investigated. Some structural characterizations, a pumping lemma and the closure properties of the resulting class of languages are established. Finally, some applications in the theory of L systems of finite index are provided.

On locally graded barely transitive groups

2013 ◽  
Vol 11 (7) ◽  
Author(s):  
Cansu Betin ◽  
Mahmut Kuzucuoğlu
Keyword(s):  
Finite Index ◽  
Cyclic Subgroup ◽  
Transitive Group ◽  
Transitive Groups ◽  
Point Stabilizer

AbstractWe show that a barely transitive group is totally imprimitive if and only if it is locally graded. Moreover, we obtain the description of a barely transitive group G for the case G has a cyclic subgroup 〈x〉 which intersects non-trivially with all subgroups and for the case a point stabilizer H of G has a subgroup H 1 of finite index in H satisfying the identity χ(H 1) = 1, where χ is a multi-linear commutator of weight w.

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