scholarly journals GROUPS OF FIBONACCI TYPE REVISITED

2012 ◽  
Vol 22 (08) ◽  
pp. 1240002 ◽  
Author(s):  
GERALD WILLIAMS
Keyword(s):  
Finite Groups ◽  
Oriented Graph ◽  
New Method ◽  
Small Cancellation ◽  
Algebraic Properties ◽  
Survey Results ◽  
Graph Groups

This article concerns a class of groups of Fibonacci type introduced by Johnson and Mawdesley that includes Conway's Fibonacci groups, the Sieradski groups, and the Gilbert–Howie groups. This class of groups provides an interesting focus for developing the theory of cyclically presented groups and, following questions by Bardakov and Vesnin and by Cavicchioli, Hegenbarth, and Repovš, they have enjoyed renewed interest in recent years. We survey results concerning their algebraic properties, such as isomorphisms within the class, the classification of the finite groups, small cancellation properties, abelianizations, asphericity, connections with Labeled Oriented Graph groups, and the semigroups of Fibonacci type. Further, we present a new method of proving the classification of the finite groups that deals with all but three groups.

scholarly journals Generalized Fibonacci groups H(r,n,s) that are connected labelled oriented graph groups

2019 ◽  
Vol 22 (1) ◽  
pp. 23-39 ◽  
Author(s):  
Gerald Williams
Keyword(s):  
Cyclic Group ◽  
Betti Numbers ◽  
Oriented Graph ◽  
Complete Classification ◽  
Circulant Matrices ◽  
Fundamental Groups ◽  
Infinite Cyclic Group ◽  
Torus Knot ◽  
Graph Groups

Abstract The class of connected Labelled Oriented Graph (LOG) groups coincides with the class of fundamental groups of complements of closed, orientable 2-manifolds embedded in {S^{4}} , and so contains all knot groups. We investigate when Campbell and Robertson’s generalized Fibonacci groups {H(r,n,s)} are connected LOG groups. In doing so, we use the theory of circulant matrices to calculate the Betti numbers of their abelianizations. We give an almost complete classification of the groups {H(r,n,s)} that are connected LOG groups. All torus knot groups and the infinite cyclic group arise and we conjecture that these are the only possibilities. As a corollary we show that {H(r,n,s)} is a 2-generator knot group if and only if it is a torus knot group.

scholarly journals Towards a classification of rigid product quotient varieties of Kodaira dimension 0

2021 ◽  
Author(s):  
Ingrid Bauer ◽  
Christian Gleissner
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Structure Theorem ◽  
Complete Classification ◽  
Kodaira Dimension ◽  
The Other ◽  
Diagonal Action ◽  
Quotient Varieties ◽  
A Finite Group

AbstractIn this paper the authors study quotients of the product of elliptic curves by a rigid diagonal action of a finite group G. It is shown that only for $$G = {{\,\mathrm{He}\,}}(3), {\mathbb {Z}}_3^2$$ G = He ( 3 ) , Z 3 2 , and only for dimension $$\ge 4$$ ≥ 4 such an action can be free. A complete classification of the singular quotients in dimension 3 and the smooth quotients in dimension 4 is given. For the other finite groups a strong structure theorem for rigid quotients is proven.

Classification of finite groups according to their conjugacy class lengths

2019 ◽  
Vol 22 (1) ◽  
pp. 137-156
Author(s):  
Zeinab Foruzanfar ◽  
İsmai̇l Ş. Güloğlu ◽  
Mehdi Rezaei
Keyword(s):  
Conjugacy Class ◽  
Finite Groups

Abstract In this paper, we classify all finite groups satisfying the following property: their conjugacy class lengths are set-wise relatively prime for any six distinct classes.

scholarly journals Counting fuzzy subgroups of some finite groups by a new equivalence relation

Filomat ◽  
2019 ◽  
Vol 33 (19) ◽  
pp. 6151-6160
Author(s):  
Ardekani Kamali
Keyword(s):  
Group Theory ◽  
Finite Groups ◽  
Equivalence Relation ◽  
Theoretical Foundation ◽  
Dihedral Groups ◽  
Significant Aspect ◽  
Exact Number ◽  
Consistent Group ◽  
Fuzzy Subgroups

The study concerning the classification of the fuzzy subgroups of finite groups is a significant aspect of fuzzy group theory. In early papers, the number of distinct fuzzy subgroups of some nonabelian groups is calculated by the natural equivalence relation. In this paper, we treat to classifying fuzzy subgroups of some groups by a new equivalence relation which has a consistent group theoretical foundation. In fact, we determine exact number of fuzzy subgroups of finite non-abelian groups of order p3 and special classes of dihedral groups.

Wildcards – natural and artificial: the combination of a panel of experts and Fuzzy TOPSIS

foresight ◽  
2017 ◽  
Vol 19 (1) ◽  
pp. 15-30 ◽  
Author(s):  
Mohsen Mohammadi ◽  
Mohammad Rahim Eivazi ◽  
Jafar Sajjadi
Keyword(s):  
Design Methodology ◽  
Fuzzy Topsis ◽  
New Method ◽  
Weak Signals ◽  
Content Type ◽  
Participatory Foresight ◽  
The Future ◽  
Similar Character ◽  
Topsis Model

Purpose The purpose of this paper is threefold: to classify wildcards into three particular types sharing similar characteristics; use the Fuzzy TOPSIS as a new method in foresight to turn qualitative ideas into quantitative ones; and apply a combination of Fuzzy TOPSIS and a panel of experts to prioritize weak signals. Design/methodology/approach In this paper, the authors classify wildcards into three particular types which share similar character: natural wildcards, artificial wildcards (Degree 1) and artificial wildcards (Degree 2). Wildcards point to unexpected and surprising events including important results that can form watershed in the development of a specific trend. In addition, the authors present a Fuzzy TOPSIS model which can be used in various cases to prioritize a number of weak signals and put them in order, so that the most important ones are likely to yield the wildcard in the future Findings The authors presented a classification of wildcards with the same characteristics being natural wildcards, artificial wildcards (Degree 1) and artificial wildcards (Degree 2). The authors also prioritized the weak signals to deal with the most important ones and take appropriate action in advance so as to minimize possible damages and maximize the benefits of potential wildcards in an uncertain environment. Originality/value In this paper, the authors report on the prioritizing of weak signals by applying Fuzzy TOPSIS and classify wildcards. This is significant because, by identifying the most important weak signals, appropriate actions can be taken in the future if necessary. The paper should be of interest to readers in the area of participatory foresight.

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