scholarly journals A new characterization of some alternating and symmetric groups

2003 ◽  
Vol 2003 (45) ◽  
pp. 2863-2872 ◽  
Author(s):  
Amir Khosravi ◽  
Behrooz Khosravi
Keyword(s):  
Prime Number ◽  
Finite Groups ◽  
Symmetric Groups ◽  
Simple Groups ◽  
Alternating And Symmetric Groups ◽  
Order Components

We suppose thatp=2α3β+1, whereα≥1, β≥0, andp≥7is a prime number. Then we prove that the simple groupsAn, wheren=p,p+1, orp+2, and finite groupsSn, wheren=p,p+1, are also uniquely determined by their order components. As corollaries of these results, the validity of a conjecture of J. G. Thompson and a conjecture of Shi and Bi (1990) both onAn, wheren=p,p+1, orp+2, is obtained. Also we generalize these conjectures for the groupsSn, wheren=p,p+1.

A NEW CHARACTERIZATION OF ALMOST SPORADIC GROUPS

2002 ◽  
Vol 01 (03) ◽  
pp. 267-279 ◽  
Author(s):  
AMIR KHOSRAVI ◽  
BEHROOZ KHOSRAVI
Keyword(s):  
Finite Group ◽  
Prime Graph ◽  
Simple Groups ◽  
Sporadic Groups ◽  
Sporadic Simple Groups ◽  
A Finite Group ◽  
Order Components ◽  
Positive Integers ◽  
Coprime Positive Integers

Let G be a finite group. Based on the prime graph of G, the order of G can be divided into a product of coprime positive integers. These integers are called order components of G and the set of order components is denoted by OC(G). Some non-abelian simple groups are known to be uniquely determined by their order components. In this paper we prove that almost sporadic simple groups, except Aut (J2) and Aut (McL), and the automorphism group of PSL(2, 2n) where n=2sare also uniquely determined by their order components. Also we discuss about the characterizability of Aut (PSL(2, q)). As corollaries of these results, we generalize a conjecture of J. G. Thompson and another conjecture of W. Shi and J. Bi for the groups under consideration.

OD-Characterization of alternating and symmetric groups of degree p + 5

2015 ◽  
Vol 36 (6) ◽  
pp. 1001-1010 ◽  
Author(s):  
Yanxiong Yan ◽  
Haijing Xu ◽  
Guiyun Chen
Keyword(s):  
Symmetric Groups ◽  
Alternating And Symmetric Groups

scholarly journals ALTERNATING AND SYMMETRIC GROUPS WITH EULERIAN GENERATING GRAPH

2017 ◽  
Vol 5 ◽  
Author(s):  
ANDREA LUCCHINI ◽  
CLAUDE MARION
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Prime Number ◽  
Symmetric Groups ◽  
Alternating Group ◽  
Generating Graph ◽  
Alternating And Symmetric Groups ◽  
A Finite Group

Given a finite group $G$, the generating graph $\unicode[STIX]{x1D6E4}(G)$ of $G$ has as vertices the (nontrivial) elements of $G$ and two vertices are adjacent if and only if they are distinct and generate $G$ as group elements. In this paper we investigate properties about the degrees of the vertices of $\unicode[STIX]{x1D6E4}(G)$ when $G$ is an alternating group or a symmetric group of degree $n$. In particular, we determine the vertices of $\unicode[STIX]{x1D6E4}(G)$ having even degree and show that $\unicode[STIX]{x1D6E4}(G)$ is Eulerian if and only if $n\geqslant 3$ and $n$ and $n-1$ are not equal to a prime number congruent to 3 modulo 4.

Natural constructions of the Mathieu groups

1989 ◽  
Vol 106 (3) ◽  
pp. 423-429 ◽  
Author(s):  
R. T. Curtis
Keyword(s):  
Symmetric Groups ◽  
The Other ◽  
Simple Groups ◽  
Alternating Group ◽  
Transitive Permutation Group ◽  
French Mathematician ◽  
Exceptional Groups ◽  
Transitive Groups ◽  
Mathieu Groups ◽  
Alternating And Symmetric Groups

In the second half of the last century the French mathematician Emil Mathieu discovered two quintuply transitive permutation groups, now labelled M12 and M24, acting on twelve and twenty-four letters respectively. With the classification of finite simple groups complete we now know that any other quintuply transitive permutation group, on any number of letters, must contain the corresponding alternating group. Indeed, the only quadruply transitive groups, other than the alternating and symmetric groups, are the point stabilizers in M12 and M24, which are denoted by M11 and M23 respectively. To put it another way, the study of multiply (≥ 4-fold) transitive groups now means the study of the symmetric groups and the Mathieu groups. Apart from their beauty and interest in their own right the Mathieu groups are involved in many of the other sporadic simple groups: see ([2], p. 238). Thus a detailed understanding of the other exceptional groups necessitates an intimate knowledge of M12 and M24.

scholarly journals On maximal subformations of n-multiple Ω-foliated formations of finite groups

2021 ◽  
Vol 21 (1) ◽  
pp. 15-25
Author(s):  
Marina M. Sorokina ◽  
◽  
Seraphim P. Maksakov ◽  
Keyword(s):  
Positive Integer ◽  
General Theory ◽  
Finite Groups ◽  
Central Place ◽  
Class I ◽  
Simple Groups ◽  
Subdirect Products ◽  
Maximal Chains

Only finite groups are considered in the article. Among the classes of groups the central place is occupied by classes closed regarding homomorphic images and subdirect products which are called formations. We study Ω-foliateded formations constructed by V. A. Vedernikov in 1999 where Ω is a nonempty subclass of the class I of all simple groups. Ω-Foliated formations are defined by two functions — an Ω-satellite f : Ω ∪ {Ω 0} → {formations} and a direction ϕ : I → {nonempty Fitting formations}. The conception of multiple locality introduced by A. N. Skiba in 1987 for formations and further developed for many other classes of groups, as applied to Ω-foliated formations is as follows: every formation is considered to be 0-multiple Ω-foliated with a direction ϕ; an Ω-foliated formation with a direction ϕ is called an n-multiple Ω-foliated formation where n is a positive integer if it has such an Ω-satellite all nonempty values of which are (n − 1)-multiple Ω-foliated formations with the direction ϕ. The aim of this work is to study the properties of maximal n-multiple Ω-foliated subformations of a given n-multiple Ω-foliated formation. We use classical methods of the theory of groups, of the theory of classes of groups, as well as methods of the general theory of lattices. In the paper we have established the existence of maximal n-multiple Ω-foliated subformations for the formations with certain properties, we have obtained the characterization of the formation ΦnΩϕ (F) which is the intersection of all maximal n-multiple Ω-foliated subformations of the formation F, and we have revealed the relation between a maximal inner Ω-satellite of 1-multiple Ω-foliated formation and a maximal inner Ω-satellite of its maximal 1-multiple Ω-foliated subformation. The results will be useful in studying the inner structure of formations of finite groups, in particular, in studying the maximal chains of subformations and in establishing the lattice properties of formations.

scholarly journals Recognition of the Simple Groups 2D8((2n)2)

2019 ◽  
Vol 57 (2) ◽  
pp. 53-60
Author(s):  
Behnam Ebrahimzadeh
Keyword(s):  
Prime Number ◽  
Finite Groups ◽  
Specific Property ◽  
Element Order ◽  
Simple Groups ◽  
Element Orders

Abstract One of the important problems in finite groups theory is group characterization by specific property. Properties, such as element orders, set of elements with the same order, the largest element order, etc. In this paper, we prove that the simple groups 2 D 8((2 n )2)where, 28 n + 1 is a prime number are uniquely determined by its order and the largest elements order.

Prosolvability criteria and properties of the prosolvable radical via Sylow sequences

2016 ◽  
Vol 19 (3) ◽  
Author(s):  
Wolfgang Herfort ◽  
Dan Levy
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Simple Groups ◽  
Profinite Groups ◽  
Sylow Subgroups ◽  
Solvability Criterion ◽  
A Finite Group

AbstractWe extend a finite group solvability criterion of J. G. Thompson, based on his classification of finite minimal simple groups, to a prosolvability criterion. Moreover, we generalize to the profinite setting subsequent developments of Thompson's criterion by G. Kaplan and the second author, which recast it in terms of properties of sequences of Sylow subgroups and their products. This generalization also encompasses a possible characterization of the prosolvable radical whose scope of validity is still open even for finite groups. We prove that if this characterization is valid for finite groups, then it carries through to profinite groups.

Finite Groups with a System of Nilpotent Subgroups

2005 ◽  
Vol 12 (02) ◽  
pp. 199-204
Author(s):  
Shirong Li ◽  
Rex S. Dark
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
A Finite Group

Let G be a finite group and p an odd prime. Let [Formula: see text] be the set of proper subgroups M of G with |G:M| not a prime power and |G:M|p=1. In this paper, we investigate the structure of G if every member of [Formula: see text] is nilpotent. In particular, a new characterization of PSL(2,7) is obtained. The proof of the theorem depends on the classification of finite simple groups.

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