scholarly journals A Note on the Normal Index and thec-Section of Maximal Subgroups of a Finite Group

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Na Tang ◽  
Xianhua Li
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Chief Factor ◽  
Maximal Subgroups ◽  
A Finite Group

LetMbe a maximal subgroup of finite groupG. For each chief factorH/KofGsuch thatK≤MandG=MH, we called the order ofH/Kthe normal index ofMandM∩H/Ka section ofMinG. Using the concepts of normal index andc-section, we obtain some new characterizations ofp-solvable, 2-supersolvable, andp-nilpotent.

scholarly journals On the σ-Length of Maximal Subgroups of Finite σ-Soluble Groups

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2165
Author(s):  
Abd El-Rahman Heliel ◽  
Mohammed Al-Shomrani ◽  
Adolfo Ballester-Bolinches
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Direct Product ◽  
Prime Numbers ◽  
Chief Factor ◽  
Maximal Subgroups ◽  
Soluble Groups ◽  
Nilpotent Length ◽  
A Finite Group ◽  
Primary Groups

Let σ={σi:i∈I} be a partition of the set P of all prime numbers and let G be a finite group. We say that G is σ-primary if all the prime factors of |G| belong to the same member of σ. G is said to be σ-soluble if every chief factor of G is σ-primary, and G is σ-nilpotent if it is a direct product of σ-primary groups. It is known that G has a largest normal σ-nilpotent subgroup which is denoted by Fσ(G). Let n be a non-negative integer. The n-term of the σ-Fitting series of G is defined inductively by F0(G)=1, and Fn+1(G)/Fn(G)=Fσ(G/Fn(G)). If G is σ-soluble, there exists a smallest n such that Fn(G)=G. This number n is called the σ-nilpotent length of G and it is denoted by lσ(G). If F is a subgroup-closed saturated formation, we define the σ-F-lengthnσ(G,F) of G as the σ-nilpotent length of the F-residual GF of G. The main result of the paper shows that if A is a maximal subgroup of G and G is a σ-soluble, then nσ(A,F)=nσ(G,F)−i for some i∈{0,1,2}.

On the 𝔉-hypercenter and the intersection of 𝔉-maximal subgroups of a finite group

2018 ◽  
Vol 21 (3) ◽  
pp. 463-473
Author(s):  
Viachaslau I. Murashka
Keyword(s):  
Finite Group ◽  
Nilpotent Groups ◽  
Chief Factor ◽  
Maximal Subgroups ◽  
Inner Automorphism ◽  
Formula Group ◽  
Saturated Formations ◽  
A Finite Group

Abstract Let {\mathfrak{X}} be a class of groups. A subgroup U of a group G is called {\mathfrak{X}} -maximal in G provided that (a) {U\in\mathfrak{X}} , and (b) if {U\leq V\leq G} and {V\in\mathfrak{X}} , then {U=V} . A chief factor {H/K} of G is called {\mathfrak{X}} -eccentric in G provided {(H/K)\rtimes G/C_{G}(H/K)\not\in\mathfrak{X}} . A group G is called a quasi- {\mathfrak{X}} -group if for every {\mathfrak{X}} -eccentric chief factor {H/K} and every {x\in G} , x induces an inner automorphism on {H/K} . We use {\mathfrak{X}^{*}} to denote the class of all quasi- {\mathfrak{X}} -groups. In this paper we describe all hereditary saturated formations {\mathfrak{F}} containing all nilpotent groups such that the {\mathfrak{F}^{*}} -hypercenter of G coincides with the intersection of all {\mathfrak{F}^{*}} -maximal subgroups of G for every group G.

scholarly journals Finite Groups with All Maximal Subgroups of Prime or Prime Square Index

1964 ◽  
Vol 16 ◽  
pp. 435-442 ◽  
Author(s):  
Joseph Kohler
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Prime Order ◽  
Maximal Subgroups ◽  
Odd Order ◽  
Order Case ◽  
Chief Factors ◽  
Even Order ◽  
A Finite Group

In this paper finite groups with the property M, that every maximal subgroup has prime or prime square index, are investigated. A short but ingenious argument was given by P. Hall which showed that such groups are solvable.B. Huppert showed that a finite group with the property M, that every maximal subgroup has prime index, is supersolvable, i.e. the chief factors are of prime order. We prove here, as a corollary of a more precise result, that if G has property M and is of odd order, then the chief factors of G are of prime or prime square order. The even-order case is different. For every odd prime p and positive integer m we shall construct a group of order 2apb with property M which has a chief factor of order larger than m.

scholarly journals A solvability condition for finite groups with nilpotent maximal subgroups

1970 ◽  
Vol 3 (2) ◽  
pp. 273-276
Author(s):  
John Randolph
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Solvability Condition ◽  
Maximal Subgroups ◽  
A Finite Group ◽  
Nilpotent Maximal Subgroup

Let G be a finite group with a nilpotent maximal subgroup S and let P denote the 2-Sylow subgroup of S. It is shown that if P ∩ Q is a normal subgroup of P for any 2-Sylow subgroup Q of G, then G is solvable.

On Prime Spectrum of Maximal Subgroups in Finite Groups

2018 ◽  
Vol 25 (04) ◽  
pp. 579-584
Author(s):  
Chi Zhang ◽  
Wenbin Guo ◽  
Natalia V. Maslova ◽  
Danila O. Revin
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Maximal Subgroups ◽  
Prime Spectrum ◽  
Prime Divisors ◽  
A Finite Group ◽  
Theoretical Results

For a positive integer n, we denote by π(n) the set of all prime divisors of n. For a finite group G, the set [Formula: see text] is called the prime spectrum of G. Let [Formula: see text] mean that M is a maximal subgroup of G. We put [Formula: see text] and [Formula: see text]. In this notice, using well-known number-theoretical results, we present a number of examples to show that both K(G) and k(G) are unbounded in general. This implies that the problem “Are k(G) and K(G) bounded by some constant k?”, raised by Monakhov and Skiba in 2016, is solved in the negative.

On Mutually m-permutable Products of Smooth Groups

2014 ◽  
Vol 57 (2) ◽  
pp. 277-282 ◽  
Author(s):  
A. M. Elkholy ◽  
M. H. Abd El-Latif
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Maximal Subgroups ◽  
A Finite Group

AbstractLet G be a finite group and H, K two subgroups of G. A group G is said to be a mutually m-permutable product of H and K if G = HK and every maximal subgroup of H permutes with K and every maximal subgroup of K permutes with H. In this paper, we investigate the structure of a finite group that is a mutually m-permutable product of two subgroups under the assumption that its maximal subgroups are totally smooth.

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.

Finite groups with given non-nilpotent maximal subgroups of prime index

2019 ◽  
Vol 18 (05) ◽  
pp. 1950087
Author(s):  
Xiaolan Yi ◽  
Shiyang Jiang ◽  
S. F. Kamornikov
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Maximal Subgroups ◽  
Subgroup Structure ◽  
A Finite Group ◽  
Nilpotent Maximal Subgroup

The subgroup structure of a finite group, under the assumption that its every non-nilpotent maximal subgroup has prime index, is studied in the paper.

An Extension of a Theorem of Janko on Finite Groups with Nilpotent Maximal Subgroups

1971 ◽  
Vol 23 (3) ◽  
pp. 550-552
Author(s):  
John W. Randolph
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Maximal Subgroups ◽  
A Finite Group ◽  
Nilpotent Maximal Subgroup

Throughout this paper G will denote a finite group containing a nilpotent maximal subgroup S and P will denote the Sylow 2-subgroup of S. The largest subgroup of S normal in G will be designated by core (S) and the largest solvable normal subgroup of G by rad(G). All other notation is standard.Thompson [6] has shown that if P = 1 then G is solvable. Janko [3] then observed that G is solvable if P is abelian, a condition subsequently weakened by him [4] to the assumption that the class of P is ≦ 2 . Our purpose is to demonstrate the sufficiency of a still weaker assumption about P.

scholarly journals c-SECTIONS, SOLVABILITY AND LARGE SUBGROUPS OF FINITE GROUPS

2012 ◽  
Vol 86 (2) ◽  
pp. 291-302
Author(s):  
BARBARA BAUMEISTER ◽  
GIL KAPLAN
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Existence Result ◽  
Composition Factor ◽  
Maximal Subgroups ◽  
Fundamental Result ◽  
Equivalent Condition ◽  
Sporadic Group ◽  
A Finite Group

Abstractc-Sections of maximal subgroups in a finite group and their relation to solvability have been extensively researched in recent years. A fundamental result due to Wang [‘C-normality of groups and its properties’, J. Algebra 180 (1998), 954–965] is that a finite group is solvable if and only if the c-sections of all its maximal subgroups are trivial. In this paper we prove that if for each maximal subgroup of a finite group G, the corresponding c-section order is smaller than the index of the maximal subgroup, then each composition factor of G is either cyclic or isomorphic to the O’Nan sporadic group (the converse does not hold). Furthermore, by a certain ‘refining’ of the latter theorem we obtain an equivalent condition for solvability. Finally, we provide an existence result for large subgroups in the sense of Lev [‘On large subgroups of finite groups’ J. Algebra 152 (1992), 434–438].

Sign in / Sign up

Export Citation Format

Share Document