Classification of finite groups by the number of element centralizers

2010 ◽  
pp. 149-157
Author(s):  
Ali Reza Ashrafi ◽  
Bijan Taeri
Keyword(s):  
Finite Groups

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.

Variations on results on orders of products in finite groups

2020 ◽  
pp. 1-7
Author(s):  
Juan Martínez ◽  
Alexander Moretó
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Simple Groups ◽  
Prime Power Order ◽  
Finite Simple Groups ◽  
Prime Divisors ◽  
Element Orders ◽  
Hall Subgroups ◽  
A Finite Group

In 2014, Baumslag and Wiegold proved that a finite group G is nilpotent if and only if o(xy) = o(x)o(y) for every x, y ∈ G with (o(x), o(y)) = 1. This has led to a number of results that characterize the nilpotence of a group (or the existence of nilpotent Hall subgroups, or the existence of normal Hall subgroups) in terms of prime divisors of element orders. Here, we look at these results with a new twist. The first of our main results asserts that G is nilpotent if and only if o(xy) ⩽ o(x)o(y) for every x, y ∈ G of prime power order with (o(x), o(y)) = 1. As an immediate consequence, we recover the Baumslag–Wiegold theorem. The proof of this result is elementary. We prove some variations of this result that depend on the classification of finite simple groups.

scholarly journals Finite groups whose all second maximal subgroups are cyclic

2017 ◽  
Vol 15 (1) ◽  
pp. 611-615 ◽  
Author(s):  
Li Ma ◽  
Wei Meng ◽  
Wanqing Ma
Keyword(s):  
Finite Groups ◽  
Complete Classification ◽  
Maximal Subgroups

Abstract In this paper, we give a complete classification of the finite groups G whose second maximal subgroups are cyclic

Classification of finite groups according to the number of conjugacy classes

1985 ◽  
Vol 51 (4) ◽  
pp. 305-338 ◽  
Author(s):  
Antonio Vera López ◽  
Juan Vera López
Keyword(s):  
Finite Groups ◽  
Conjugacy Classes

scholarly journals Classification of Finite Groups with Toroidal or Projective-Planar Permutability Graphs

2016 ◽  
Vol 44 (9) ◽  
pp. 3705-3726 ◽  
Author(s):  
R. Rajkumar ◽  
P. Devi ◽  
Andrei Gagarin
Keyword(s):  
Finite Groups

CLASSIFICATION OF FINITE GROUPS VIA THEIR BREADTH

2019 ◽  
Vol 62 (3) ◽  
pp. 544-563 ◽  
Author(s):  
HERMANN HEINEKEN ◽  
FRANCESCO G. RUSSO
Keyword(s):  
Inverse Problem ◽  
Finite Group ◽  
Finite Groups ◽  
A Finite Group

AbstractLet k be a divisor of a finite group G and Lk(G) = {x ∈ G | xk =1}. Frobenius proved that the number |Lk(G)| is always divisible by k. The following inverse problem is considered: for a given integer n, find all groups G such that max{k-1|Lk(G)| | k ∈ Div(G)} = n, where Div(G) denotes the set of all divisors of |G|. A procedure beginning with (in a sense) minimal members and deducing the remaining ones is outlined and executed for n=8.

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.

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