scholarly journals Nonsoluble length of finite groups with commutators of small order

2015 ◽  
Vol 158 (3) ◽  
pp. 487-492 ◽  
Author(s):  
Y. CONTRERAS–ROJAS ◽  
PAVEL SHUMYATSKY
Keyword(s):  
Finite Group ◽  
Direct Product ◽  
Finite Groups ◽  
Simple Groups ◽  
Group Variety ◽  
Multilinear Commutator ◽  
Normal Series ◽  
The Law ◽  
The One ◽  
Proper Group

AbstractLet p be a prime. Every finite group G has a normal series each of whose quotients either is p-soluble or is a direct product of nonabelian simple groups of orders divisible by p. The non-p-soluble length λp(G) is defined as the minimal number of non-p-soluble quotients in a series of this kind.We deal with the question whether, for a given prime p and a given proper group variety , there is a bound for the non-p-soluble length λp of finite groups whose Sylow p-subgroups belong to . Let the word w be a multilinear commutator. In this paper we answer the question in the affirmative in the case where p is odd and the variety is the one of groups satisfying the law we ≡ 1.

Finite groups having unique proper characteristic subgroups. I

1955 ◽  
Vol 51 (1) ◽  
pp. 25-36 ◽  
Author(s):  
D. R. Taunt
Keyword(s):  
Finite Group ◽  
Direct Product ◽  
Finite Groups ◽  
Simple Groups ◽  
Characteristic Subgroup ◽  
Direct Products ◽  
A Finite Group

It is well known that a characteristically-simple finite group, that is, a group having no characteristic subgroup other than itself and the identity subgroup, must be either simple or the direct product of a number of isomorphic simple groups. It was suggested to the author by Prof. Hall that finite groups possessing exactly one proper characteristic subgroup would repay attention. We shall call a finite group having a unique proper characteristic subgroup a ‘UCS group’. In the present paper we first give some results on direct products of isomorphic UCS groups, and then we consider in more detail one of the types of UCS groups which can exist, that consisting of groups whose orders are divisible by exactly two distinct primes.

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.

Finite groups with at most six vanishing conjugacy classes

2021 ◽  
pp. 2250076
Author(s):  
Sajjad M. Robati ◽  
M. R. Darafsheh
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Irreducible Character ◽  
Conjugacy Classes ◽  
Character Formula ◽  
Simple Groups ◽  
Almost Simple Groups ◽  
A Finite Group

Let [Formula: see text] be a finite group. We say that a conjugacy class of [Formula: see text] in [Formula: see text] is vanishing if there exists some irreducible character [Formula: see text] of [Formula: see text] such that [Formula: see text]. In this paper, we show that finite groups with at most six vanishing conjugacy classes are solvable or almost simple groups.

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.

Some Remarks on Groups Admitting a Fixed-Point-Free Automorphism

1968 ◽  
Vol 20 ◽  
pp. 1300-1307 ◽  
Author(s):  
Fletcher Gross
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Simple Group ◽  
Direct Product ◽  
Finite Simple Group ◽  
Identity Element ◽  
Simple Groups ◽  
Cyclic Groups ◽  
A Finite Group ◽  
Open Question

A finite group G is said to be a fixed-point-free-group (an FPF-group) if there exists an automorphism a which fixes only the identity element of G. The principal open question in connection with these groups is whether non-solvable FPF-groups exist. One of the results of the present paper is that if a Sylow p-group of the FPF-group G is the direct product of any number of mutually non-isomorphic cyclic groups, then G has a normal p-complement. As a consequence of this, the conjecture that all FPF-groups are solvable would be true if it were true that every finite simple group has a non-trivial SylowT subgroup of the kind just described. Here it should be noted that all the known simple groups satisfy this property.

On Groups with all Composition Factors Isomorphic

1966 ◽  
Vol 9 (4) ◽  
pp. 413-415 ◽  
Author(s):  
R. Bercov
Keyword(s):  
Finite Group ◽  
Direct Product ◽  
Simple Groups ◽  
Composition Series ◽  
Composition Factors

By the celebrated theorem of Jordan [3] and Hölder [2], there is associated with each finite group G a family of distinct simple groups Hi. such that every composition series of G has ni factor groups isomorphic to Hi and no others. We denote the collection of pairs (Hi, ni) by CF(G). Conversely, given k pairs (Hi, ni), we may construct by an easy direct product procedure a group G with CF(G) = { (Hi, ni) | i =1,…, k}. The composition factors, of course, do not in general determine the group.

Gradewise Properties of Subgroups and Their Applications

2017 ◽  
Vol 24 (02) ◽  
pp. 255-266
Author(s):  
Wenbin Guo ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Binary Relation ◽  
Finite Groups ◽  
Normal Series ◽  
A Finite Group

For each finite group E, let Θ(E) be a binary relation on the set of all subgroups of E. If A and B are subgroups of a finite group G, then we say that the pair (A, B) enjoys the gradewise property (resp., generalized gradewise property) Θ in G if G has a normal series Γ : 1 = G0 [Formula: see text] such that for each [Formula: see text], we have [Formula: see text] (resp., we have [Formula: see text]. Using these concepts, we obtain some new characterizations for solubility and supersolubility of finite groups and generalize some known results.

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.

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