Cyclic Groups and Cyclic Subgroups

1996 ◽  
pp. 15-20
Keyword(s):  
Cyclic Subgroups ◽  
Cyclic Groups

Sur les Sous-Groupes Purifiables des Groupes Abéliens Primaires

1990 ◽  
Vol 33 (1) ◽  
pp. 11-17 ◽  
Author(s):  
K. Benabdallah ◽  
C. Piché
Keyword(s):  
Abelian Groups ◽  
Divisible Group ◽  
Direct Sums ◽  
Cyclic Subgroups ◽  
Cyclic Groups ◽  
Direct Sum

AbstractThe class of primary abelian groups whose subsocles are purifiable is not yet completely characterized and it contains the class of direct sums of cyclic groups and torsion complete groups. In sharp constrast with this, the class of groups whose p2-bounded subgroups are purifiable consist only of those groups which are the direct sum of a bounded and a divisible group. Various tools are developed and a short application to the pure envelopes of cyclic subgroups is given in the last section.

scholarly journals On the Exponent of the NK0-Groups of Virtually Infinite Cyclic Groups

2002 ◽  
Vol 45 (2) ◽  
pp. 180-195 ◽  
Author(s):  
Francis X. Connolly ◽  
Stratos Prassidis
Keyword(s):  
Large Class ◽  
The Other ◽  
Main Difficulty ◽  
Cyclic Subgroups ◽  
Cyclic Groups ◽  
Other Hand

AbstractIt is known that the K-theory of a large class of groups can be computed from the K-theory of their virtually infinite cyclic subgroups. On the other hand, Nil-groups appear to be the obstacle in calculations involving the K-theory of the latter. The main difficulty in the calculation of Nil-groups is that they are infinitely generated when they do not vanish. We develop methods for computing the exponent of NK0-groups that appear in the calculation of the K0-groups of virtually infinite cyclic groups.

scholarly journals Classifying Spaces for the Family of Virtually Cyclic Subgroups of Braid Groups

2018 ◽  
Vol 2020 (5) ◽  
pp. 1575-1600
Author(s):  
Ramón Flores ◽  
Juan González-Meneses
Keyword(s):  
Braid Group ◽  
Braid Groups ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Cyclic Subgroups ◽  
Cyclic Groups ◽  
Pure Braid Group ◽  
Minimal Dimension ◽  
The Family

Abstract We prove that, for n ≥ 3, the minimal dimension of a model of the classifying space of the braid group $B_{n}$, and of the pure braid group $P_{n}$, with respect to the family of virtually cyclic groups is n.

scholarly journals Products of locally cyclic groups

2021 ◽  
Author(s):  
Bernhard Amberg ◽  
Yaroslav Sysak
Keyword(s):  
Finite Number ◽  
Finite Groups ◽  
Complete Classification ◽  
The Other ◽  
Supersoluble Group ◽  
Cyclic Subgroups ◽  
Cyclic Groups ◽  
Prüfer Rank ◽  
Torsion Free

AbstractWe consider groups of the form $${G} = {AB}$$ G = AB with two locally cyclic subgroups A and B. The structure of these groups is determined in the cases when A and B are both periodic or when one of them is periodic and the other is not. Together with a previous study of the case where A and B are torsion-free, this gives a complete classification of all groups that are the product of two locally cyclic subgroups. As an application, it is shown that the Prüfer rank of a periodic product of two locally cyclic subgroups does not exceed 3, and this bound is sharp. It is also proved that a product of a finite number of pairwise permutable periodic locally cyclic subgroups is a locally supersoluble group. This generalizes a well-known theorem of B. Huppert for finite groups.

scholarly journals Searching Big Data via cyclic groups

2017 ◽  
Vol 6 (2) ◽  
pp. 47-53
Author(s):  
Timur Karacay
Keyword(s):  
Big Data ◽  
Group Structure ◽  
Linear Search ◽  
Topological Groups ◽  
Cyclic Subgroups ◽  
Cyclic Groups ◽  
Dual Groups

We look up for a certain information in big data. To achieve this task we first endow the big data with a group structure and partition it to it’s cyclic subgroups. We devise a method to search the whole big data starting from the smallest subroup through the largest one. Our method eventually exhausts the whole big data. Keywords: BigData, topological groups, dual groups, linear search.

Torsion

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Finite Order ◽  
Class Group ◽  
Mapping Class ◽  
Cyclic Subgroups ◽  
Cyclic Groups ◽  
Hyperbolic Orbifold ◽  
Finite Subgroups ◽  
Mapping Classes ◽  
Bonnet Theorem ◽  
Nielsen Realization

This chapter deals with finite subgroups of the mapping class group. It first explains the distinction between finite-order mapping classes and finite-order homeomorphisms, focusing on the Nielsen realization theorem for cyclic groups and detection of torsion with the symplectic representation. It then considers the problem of finding an Euler characteristic for orbifolds, to prove a Gauss–Bonnet theorem for orbifolds, and to use these results to show that there is a universal lower bound of π‎/21 for the area of any 2-dimensional orientable hyperbolic orbifold. The chapter demonstrates that, when g is greater than or equal to 2, finite subgroups have order at most 84(g − 1) and cyclic subgroups have order at most 4g + 2. It also describes finitely many conjugacy classes of finite subgroups in Mod(S) and concludes by proving that Mod(Sɡ) is generated by finitely many elements of order 2.

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 the minimal varieties of PI-algebras graded by finite cyclic groups

2021 ◽  
pp. 1-25
Author(s):  
Marcos Antônio da Silva Pinto ◽  
Viviane Ribeiro Tomaz da Silva
Keyword(s):  
Cyclic Groups
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