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 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.

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.

scholarly journals A Characterization of Odd Order Extensions of the Finite Simple Groups PSp(4, q), G2(q),

1972 ◽  
Vol 45 ◽  
pp. 79-96 ◽  
Author(s):  
Morton E. Harris
Keyword(s):  
Positive Integer ◽  
Conjugacy Class ◽  
Symplectic Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Prime Integer ◽  
Twisted Group ◽  
Odd Order

Let p denote an odd prime integer and let q = pf where f is a positive integer. Let ℋ denote the projective symplectic group PSp(4,q), the Dickson group G2(q), or the Steinberg “triality twisted” group over a field Fq of q elements. Then ℋ is simple and the Sylow 2-subgroups of ℋ have centers of order 2 so that involutions which centralize a Sylow 2-subgroup of ℋ form a single conjugacy class of ℋ.

scholarly journals Elements of prime power order and their conjugacy classes in finite groups

2005 ◽  
Vol 78 (2) ◽  
pp. 291-295 ◽  
Author(s):  
László Héthelyi ◽  
Burkhard Külshammer
Keyword(s):  
Positive Integer ◽  
Finite Groups ◽  
Prime Power ◽  
Conjugacy Classes ◽  
Simple Groups ◽  
Prime Power Order ◽  
Finite Simple Groups

AbstractWe show that, for any positive integer k, there are only finitely many finite groups, up to isomorphism, with exactly k conjugacy classes of elements of prime power order. This generalizes a result of E. Landau from 1903. The proof of our generalization makes use of the classification of finite simple groups.

scholarly journals Characterization of novel zebrafish MHC class I U lineage genes and their haplotype

2021 ◽  
Vol 116 ◽  
pp. 103952
Author(s):  
Yasuko Honjo ◽  
Kosuke Takano ◽  
Tatsuo Ichinohe
Keyword(s):  
Mhc Class I ◽  
Class I

scholarly journals A new characterization of L2(p2)

2020 ◽  
Vol 18 (1) ◽  
pp. 907-915
Author(s):  
Zhongbi Wang ◽  
Chao Qin ◽  
Heng Lv ◽  
Yanxiong Yan ◽  
Guiyun Chen
Keyword(s):  
Positive Integer ◽  
Irreducible Character ◽  
Character Degrees ◽  
Prime Divisors

Abstract For a positive integer n and a prime p, let {n}_{p} denote the p-part of n. Let G be a group, \text{cd}(G) the set of all irreducible character degrees of G , \rho (G) the set of all prime divisors of integers in \text{cd}(G) , V(G)=\left\{{p}^{{e}_{p}(G)}|p\in \rho (G)\right\} , where {p}^{{e}_{p}(G)}=\hspace{.25em}\max \hspace{.25em}\{\chi {(1)}_{p}|\chi \in \text{Irr}(G)\}. In this article, it is proved that G\cong {L}_{2}({p}^{2}) if and only if |G|=|{L}_{2}({p}^{2})| and V(G)=V({L}_{2}({p}^{2})) .

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