scholarly journals Closed orbits in quotient systems

2016 ◽  
Vol 37 (7) ◽  
pp. 2337-2352
Author(s):  
STEFANIE ZEGOWITZ
Keyword(s):  
Growth Rate ◽  
Finite Group ◽  
Dynamical Systems ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Closed Orbits ◽  
Quotient System ◽  
A Finite Group ◽  
Topological Dynamical Systems ◽  
The Relationship

We study the relationship between pairs of topological dynamical systems $(X,T)$ and $(X^{\prime },T^{\prime })$, where $(X^{\prime },T^{\prime })$ is the quotient of $(X,T)$ under the action of a finite group $G$. We describe three phenomena concerning the behaviour of closed orbits in the quotient system, and the constraints given by these phenomena. We find upper and lower bounds for the extremal behaviour of closed orbits in the quotient system in terms of properties of $G$ and show that any growth rate in between these bounds can be achieved.

The probability that a pair of group elements is 2-Engel

2021 ◽  
pp. 2250088
Author(s):  
Raheleh Khoshtarash ◽  
Mohammad Reza R. Moghaddam ◽  
Mohammad Amin Rostamyari
Keyword(s):  
Finite Group ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
A Finite Group

In 2016, the second and third authors introduced the notion of 2-Engelizer of an element [Formula: see text] in a given group [Formula: see text] and the set of all such 2-Engelizers in [Formula: see text] denoted by [Formula: see text]. In this paper, among other results, we construct some upper and lower bounds for the probability of randomly chosen a pair of elements of a finite group to be 2-Engel.

THE PROBABILITY OF AN AUTOMORPHISM FIXING A SUBGROUP ELEMENT OF A FINITE GROUP

2011 ◽  
Vol 04 (02) ◽  
pp. 301-308 ◽  
Author(s):  
Mohammad Reza R. Moghaddam ◽  
Farshid Saeedi ◽  
Elaheh Khamseh
Keyword(s):  
Finite Group ◽  
Lower Bounds ◽  
Arbitrary Element ◽  
Upper And Lower Bounds ◽  
Probability Concept ◽  
A Finite Group

In 1975, Sherman introduced the probability of an automorphism of a finite group, which fixes an arbitrary element of the group. In this paper we introduce a new probability concept, namely the probability of an automorphism of a given finite group such that it fixes a subgroup element of the group. Among other results, we construct some upper and lower bounds for both probabilities.

scholarly journals Enumerating Finite Racks, Quandles and Kei

10.37236/3262 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Simon R. Blackburn
Keyword(s):  
Permutation Group ◽  
Lower Bounds ◽  
Binary Operation ◽  
Independent Interest ◽  
Upper And Lower Bounds ◽  
Permutation Groups ◽  
Isomorphism Classes ◽  
Operator Group ◽  
The Relationship

A rack of order $n$ is a binary operation $\vartriangleright$ on a set $X$ of cardinality $n$, such that right multiplication is an automorphism. More precisely, $(X,\vartriangleright)$ is a rack provided that the map $x\mapsto x\vartriangleright y$ is a bijection for all $y\in X$, and $(x\vartriangleright y)\vartriangleright z=(x\vartriangleright z)\vartriangleright (y\vartriangleright z)$ for all $x,y,z\in X$.The paper provides upper and lower bounds of the form $2^{cn^2}$ on the number of isomorphism classes of racks of order $n$. Similar results on the number of isomorphism classes of quandles and kei are obtained. The results of the paper are established by first showing how an arbitrary rack is related to its operator group (the permutation group on $X$ generated by the maps $x\mapsto x\vartriangleright y$ for $y\in Y$), and then applying some of the theory of permutation groups. The relationship between a rack and its operator group extends results of Joyce and of Ryder; this relationship might be of independent interest.

An answer to a conjecture on the sum of element orders

2020 ◽  
pp. 2250067
Author(s):  
Morteza Baniasad Azad ◽  
Behrooz Khosravi ◽  
Morteza Jafarpour
Keyword(s):  
Finite Group ◽  
Lower Bound ◽  
Nilpotent Group ◽  
Prime Number ◽  
Lower Bounds ◽  
Finite Groups ◽  
Open Problem ◽  
Element Orders ◽  
Equality Condition ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text], where [Formula: see text] denotes the order of [Formula: see text]. The function [Formula: see text] was introduced by Tărnăuceanu. In [M. Tărnăuceanu, Detecting structural properties of finite groups by the sum of element orders, Israel J. Math. (2020), https://doi.org/10.1007/s11856-020-2033-9 ], some lower bounds for [Formula: see text] are determined such that if [Formula: see text] is greater than each of them, then [Formula: see text] is cyclic, abelian, nilpotent, supersolvable and solvable. Also, an open problem aroused about finite groups [Formula: see text] such that [Formula: see text] is equal to the amount of each lower bound. In this paper, we give an answer to the equality condition which is a partial answer to the open problem posed by Tărnăuceanu. Also, in [M. Baniasad Azad and B. Khosravi, A criterion for p-nilpotency and p-closedness by the sum of element orders, Commun. Algebra (2020), https://doi.org/10.1080/00927872.2020.1788571 ], it is shown that: If [Formula: see text], where [Formula: see text] is a prime number, then [Formula: see text] and [Formula: see text] is cyclic. As the next result, we show that if [Formula: see text] is not a [Formula: see text]-nilpotent group and [Formula: see text], then [Formula: see text].

On the existence of traveling wave solutions and upper and lower bounds for some Fisher–Kolmogorov type equations

2014 ◽  
Vol 07 (05) ◽  
pp. 1450050 ◽  
Author(s):  
Juan Belmonte-Beitia
Keyword(s):  
Mathematical Biology ◽  
Dynamical Systems ◽  
Lower Bounds ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Systems Approach ◽  
Traveling Wave Solutions ◽  
Upper And Lower Bounds ◽  
Kolmogorov Type ◽  
Wave Solutions

In this paper, we use a dynamical systems approach to prove the existence of traveling waves solutions for the Fisher–Kolmogorov density-dependent equation. Moreover, we prove the existence of upper and lower bounds for these traveling wave solutions found previously. Finally, we present a particular example which has several applications in the mathematical biology field.

Characteristics of GWrSn

1976 ◽  
Vol 79 (3) ◽  
pp. 433-441
Author(s):  
A. G. Williams
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Wreath Product ◽  
Generating Functions ◽  
Block Structure ◽  
Clifford Theory ◽  
A Finite Group ◽  
The Relationship

The ‘characteristics’ of the wreath product GWrSn, where G is a finite group, are certain polynomials (to be defined in section 2) which are generating functions for the simple characters of GWrSn. Schur (8) first used characteristics of the symmetric group. Specht (9) defined characteristics for GWrSn and found a relation between the characteristics of GWrSn and those of Sn which determined the simple characters of GWrSn. The object of this paper is to describe the p-block structure of GWrSn in the case where p is not a factor of the order of G. We use the relationship between the characteristics of GWrSn and those of Sn, which we deduce from a knowledge of the simple characters of GWrSn (these can be determined, independently of Specht's work, by using Clifford theory).

Growth rates of bending KdV solitons

1982 ◽  
Vol 28 (3) ◽  
pp. 469-484 ◽  
Author(s):  
E. W. Laedke ◽  
K. H. Spatschek
Keyword(s):  
Growth Rate ◽  
Lower Bounds ◽  
Acoustic Waves ◽  
Variational Principles ◽  
Soliton Solutions ◽  
Type Equation ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Upper And Lower Bounds ◽  
Ion Acoustic Waves

Nonlinear ion-acoustic waves in magnetized plasmas are investigated. In strong magnetic fields they can be described by a Korteweg-de Vries (KdV) type equation. It is shown here that these plane soliton solutions become unstable with respect to bending distortions. Variational principles are derived for the maximum growth rate γ as a function of the transverse wavenumber k of the perturbations. Since the variational principles are formulated in complementary form, the numerical evaluation yields upper and lower bounds for γ. Choosing appropriate test functions and increasing the accuracy of the computations we find very close upper and lower bounds for the γ(k) curve. The results show that the growth rate peaks at a certain value of k and a cut-off kc exists. In the region where the γ(k) curve was not predicted numerically with high accuracy, i.e. near the cut-off, we find very precise analytical estimates. These findings are compared with previous results. For k≥kc, stability with respect to transverse perturbations is proved.

On the Intersection of the Normalizers of the Nilpotent Residuals of All Subgroups of a Finite Group

2013 ◽  
Vol 20 (02) ◽  
pp. 349-360 ◽  
Author(s):  
Lü Gong ◽  
Xiuyun Guo
Keyword(s):  
Finite Group ◽  
Characteristic Subgroup ◽  
A Finite Group ◽  
The Relationship

In this paper, a characteristic subgroup [Formula: see text] of a finite group G is defined, which is the intersection of the normalizers of the nilpotent residuals of all subgroups of G, and the properties of [Formula: see text] and the relationship between [Formula: see text] and the group G are investigated.

Weak Cayley tables and generalized centralizer rings of finite groups

2012 ◽  
Vol 153 (2) ◽  
pp. 281-318 ◽  
Author(s):  
STEPHEN P. HUMPHRIES ◽  
EMMA L. RODE
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Cayley Table ◽  
A Finite Group ◽  
Finite Conjugacy ◽  
Relationship Of ◽  
Cayley Tables ◽  
The Relationship

AbstractFor a finite group G we study certain rings (k)G called k-S-rings, one for each k ≥ 1, where (1)G is the centraliser ring Z(ℂG) of G. These rings have the property that (k+1)G determines (k)G for all k ≥ 1. We study the relationship of (2)G with the weak Cayley table of G. We show that (2)G and the weak Cayley table together determine the sizes of the derived factors of G (noting that a result of Mattarei shows that (1)G = Z(ℂG) does not). We also show that (4)G determines G for any group G with finite conjugacy classes, thus giving an answer to a question of Brauer. We give a criteria for two groups to have the same 2-S-ring and a result guaranteeing that two groups have the same weak Cayley table. Using these results we find a pair of groups of order 512 that have the same weak Cayley table, are a Brauer pair, and have the same 2-S-ring.

scholarly journals Remarks on upper and lower bounds formatching sequencibility of graphs

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2091-2099
Author(s):  
Shuya Chiba ◽  
Yuji Nakano
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Edge Coloring ◽  
Hamilton Cycle ◽  
Upper And Lower Bounds ◽  
Linear Ordering ◽  
The Relationship

In 2008, Alspach [The Wonderful Walecki Construction, Bull. Inst. Combin. Appl. 52 (2008) 7-20] defined the matching sequencibility of a graph G to be the largest integer k such that there exists a linear ordering of its edges so that every k consecutive edges in the linear ordering form a matching of G, which is denoted by ms(G). In this paper, we show that every graph G of size q and maximum degree ? satisfies 1/2?q/?+1? ? ms(G) ? ?q?1/??1? by using the edge-coloring of G, and we also improve this lower bound for some particular graphs. We further discuss the relationship between the matching sequencibility and a conjecture of Seymour about the existence of the kth power of a Hamilton cycle.

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