scholarly journals Some properties of various graphs associated with finite groups

2021 ◽  
Vol 31 (2) ◽  
pp. 195-211
Author(s):  
X. Y. Chen ◽  
◽  
A. R. Moghaddamfar ◽  
M. Zohourattar ◽  
◽  
...  
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Explicit Formula ◽  
Finite Groups ◽  
Simple Groups ◽  
Power Graph ◽  
Tree Number ◽  
A Finite Group ◽  
Free Decomposition

In this paper we investigate some properties of the power graph and commuting graph associated with a finite group, using their tree-numbers. Among other things, it is shown that the simple group L2(7) can be characterized through the tree-number of its power graph. Moreover, the classification of groups with power-free decomposition is presented. Finally, we obtain an explicit formula concerning the tree-number of commuting graphs associated with the Suzuki simple groups.

scholarly journals A Study About One Generation of Finite Simple Groups and Finite Groups

2021 ◽  
Vol 13 (3) ◽  
pp. 59
Author(s):  
Nader Taffach
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Prime Numbers ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
A Finite Group

In this paper, we study the problem of how a finite group can be generated by some subgroups. In order to the finite simple groups, we show that any finite non-abelian simple group can be generated by two Sylow p1 - and p_2 -subgroups, where p_1  and p_2  are two different primes. We also show that for a given different prime numbers p  and q , any finite group can be generated by a Sylow p -subgroup and a q -subgroup.

On mutually permutable products of generalized supersoluble subgroups of finite groups

2016 ◽  
Vol 09 (03) ◽  
pp. 1650054
Author(s):  
E. N. Myslovets
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Paper Properties ◽  
Supersoluble Groups ◽  
A Finite Group ◽  
Mutually Permutable Products ◽  
Composition Factors

Let [Formula: see text] be a class of finite simple groups. We say that a finite group [Formula: see text] is a [Formula: see text]-group if all composition factors of [Formula: see text] are contained in [Formula: see text]. A group [Formula: see text] is called [Formula: see text]-supersoluble if every chief [Formula: see text]-factor of [Formula: see text] is a simple group. In this paper, properties of mutually permutable products of [Formula: see text]-supersoluble finite groups are studied. Some earlier results on mutually permutable products of [Formula: see text]-supersoluble groups (SC-groups) appear as particular cases.

scholarly journals Recognition of symplectic and orthogonal groups of small dimensions by spectrum

2019 ◽  
Vol 18 (12) ◽  
pp. 1950230
Author(s):  
Mariya A. Grechkoseeva ◽  
Andrey V. Vasil’ev ◽  
Mariya A. Zvezdina
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Simple Groups ◽  
Orthogonal Groups ◽  
A Finite Group ◽  
Field Automorphism ◽  
Orders Of Elements

We refer to the set of the orders of elements of a finite group as its spectrum and say that finite groups are isospectral if their spectra coincide. In this paper, we determine all finite groups isospectral to the simple groups [Formula: see text], [Formula: see text], and [Formula: see text]. In particular, we prove that with just four exceptions, every such finite group is an extension of the initial simple group by a (possibly trivial) field automorphism.

A Class of Finite Groups with Abelian Sylow p-Subgroups

2006 ◽  
Vol 13 (03) ◽  
pp. 471-480
Author(s):  
Zhikai Zhang
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Defect Groups ◽  
Abelian Defect Groups ◽  
A Finite Group

In this paper, we first determine the structure of the Sylow p-subgroup P of a finite group G containing no elements of order 2p (p > 2), and then show that the Broué Abelian Defect Groups Conjecture is true for the principal p-block of G. The result depends on the classification of finite simple groups.

On residuals of finite groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Stefanos Aivazidis ◽  
Thomas Müller
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Nilpotent Groups ◽  
Simple Groups ◽  
Fitting Subgroup ◽  
Soluble Groups ◽  
A Finite Group ◽  
Fitting Formation ◽  
Original Theorem

Abstract Theorem C in [S. Dolfi, M. Herzog, G. Kaplan and A. Lev, The size of the solvable residual in finite groups, Groups Geom. Dyn. 1 (2007), 4, 401–407] asserts that, in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order and that the inequality is sharp. Inspired by this result and some of the arguments in the above article, we establish the following generalisation: if 𝔛 is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and X ¯ \overline{\mathfrak{X}} is the extension-closure of 𝔛, then there exists an (explicitly known and optimal) constant 𝛾 depending only on 𝔛 such that, for all non-trivial finite groups 𝐺 with trivial 𝔛-radical, | G X ¯ | > | G | γ \lvert G^{\overline{\mathfrak{X}}}\rvert>\lvert G\rvert^{\gamma} , where G X ¯ G^{\overline{\mathfrak{X}}} is the X ¯ \overline{\mathfrak{X}} -residual of 𝐺. When X = N \mathfrak{X}=\mathfrak{N} , the class of finite nilpotent groups, it follows that X ¯ = S \overline{\mathfrak{X}}=\mathfrak{S} , the class of finite soluble groups; thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J. G. Thompson’s classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations 𝔛 of full characteristic such that S ⊂ X ¯ ⊂ E \mathfrak{S}\subset\overline{\mathfrak{X}}\subset\mathfrak{E} , where 𝔈 denotes the class of all finite groups, thus providing applications of our main result beyond the reach of the above theorem.

scholarly journals GENERATING MAXIMAL SUBGROUPS OF FINITE ALMOST SIMPLE GROUPS

2020 ◽  
Vol 8 ◽  
Author(s):  
ANDREA LUCCHINI ◽  
CLAUDE MARION ◽  
GARETH TRACEY
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Simple Group ◽  
Finite Groups ◽  
Upper Bounds ◽  
Maximal Subgroups ◽  
Simple Groups ◽  
Almost Simple Group ◽  
Almost Simple Groups ◽  
A Finite Group

For a finite group $G$ , let $d(G)$ denote the minimal number of elements required to generate $G$ . In this paper, we prove sharp upper bounds on $d(H)$ whenever $H$ is a maximal subgroup of a finite almost simple group. In particular, we show that $d(H)\leqslant 5$ and that $d(H)\geqslant 4$ if and only if $H$ occurs in a known list. This improves a result of Burness, Liebeck and Shalev. The method involves the theory of crowns in finite groups.

Constructing Representations of Finite Simple Groups and Covers

2006 ◽  
Vol 58 (1) ◽  
pp. 23-38 ◽  
Author(s):  
Vahid Dabbaghian-Abdoly
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Irreducible Character ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Simple Method ◽  
Covering Group ◽  
Representations Of Finite Groups ◽  
A Finite Group

AbstractLet G be a finite group and χ be an irreducible character of G. An efficient and simple method to construct representations of finite groups is applicable whenever G has a subgroup H such that χH has a linear constituent with multiplicity 1. In this paper we show (with a few exceptions) that if G is a simple group or a covering group of a simple group and χ is an irreducible character of G of degree less than 32, then there exists a subgroup H (often a Sylow subgroup) of G such that χH has a linear constituent with multiplicity 1.

ON THE RECOGNITION OF THE SIMPLE GROUPS L7(3) AND L8(3) BY THE SPECTRUM

2008 ◽  
Vol 18 (05) ◽  
pp. 925-933 ◽  
Author(s):  
M. R. DARAFSHEH
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Finite Groups ◽  
Simple Groups ◽  
A Finite Group

In this paper we will prove that up to isomorphism there are two finite groups with the same spectrum as that of the simple group L7(3). For the group L8(3) we will prove that if G is a finite group with the same spectrum as that of L8(3), then G is isomorphic to the group L8(3).

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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