On the generalized norm of a finite group

2015 ◽  
Vol 15 (01) ◽  
pp. 1650008 ◽  
Author(s):  
Lü Gong ◽  
Libo Zhao ◽  
Xiuyun Guo
Keyword(s):  
Finite Group ◽  
A Finite Group ◽  
Subnormal Subgroups

The main aim of this paper is to investigate two characteristic subgroups ω𝒜(G) and θ𝒜(G) of a finite group G, which are defined as the intersections of the normalizers of derived subgroups of subnormal and non-subnormal subgroups of G respectively. Our main theory improve and extend some earlier results.

Generalized Fitting subgroup of a group of finite Morley rank

1991 ◽  
Vol 56 (4) ◽  
pp. 1391-1399 ◽  
Author(s):  
Ali Nesin
Keyword(s):  
Finite Group ◽  
Morley Rank ◽  
Fitting Subgroup ◽  
Generalized Fitting Subgroup ◽  
Finite Morley Rank ◽  
The Generalized Fitting Subgroup ◽  
A Finite Group ◽  
Subnormal Subgroups

AbstractWe define a characteristic and definable subgroup F*(G) of any group G of finite Morley rank that behaves very much like the generalized Fitting subgroup of a finite group. We also prove that semisimple subnormal subgroups of G are all definable and that there are finitely many of them.

Finite groups with short nonnormal chains

1974 ◽  
Vol 18 (1) ◽  
pp. 111-118
Author(s):  
Armond E. Spencer
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
A Finite Group ◽  
Subnormal Subgroups

This note is a continuation of the author's work [6], describing the structure of a finite group given some information about the distribution of the subnormal subgroups in the lattice of all subgoups.DEFINITION. An upper chain of length n in the finite group G is a sequence of subgroups of G; G = Go > G1 > … > Gn, such that for each i, Gi is a maximal subgroup of Gi-1. Let h(G) = n if every upper chain in G of length n contains a proper ( ≠ G) subnormal entry, and there is at least one upper chain in G of length (n – 1) which contains no proper subnormal entry.

scholarly journals ON GENERALISED PRONORMAL SUBGROUPS OF FINITE GROUPS

2014 ◽  
Vol 56 (3) ◽  
pp. 691-703 ◽  
Author(s):  
A. BALLESTER-BOLINCHES ◽  
J. C. BEIDLEMAN ◽  
A. D. FELDMAN ◽  
M. F. RAGLAND
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Subnormal Subgroup ◽  
Normal Subgroups ◽  
Transitive Relation ◽  
Formula Group ◽  
Permutable Subgroups ◽  
Sylow Permutable ◽  
A Finite Group ◽  
Subnormal Subgroups

AbstractFor a formation $\mathfrak F$, a subgroup M of a finite group G is said to be $\mathfrak F$-pronormal in G if for each g ∈ G, there exists x ∈ 〈U,Ug〉$\mathfrak F$ such that Ux = Ug. Let f be a subgroup embedding functor such that f(G) contains the set of normal subgroups of G and is contained in the set of Sylow-permutable subgroups of G for every finite group G. Given such an f, let fT denote the class of finite groups in which f(G) is the set of subnormal subgroups of G; this is the class of all finite groups G in which to be in f(G) is a transitive relation in G. A subgroup M of a finite group G is said to be $\mathfrak F$-normal in G if G/CoreG(M) belongs to $\mathfrak F$. A subgroup U of a finite group G is called K-$\mathfrak F$-subnormal in G if either U = G or there exist subgroups U = U0 ≤ U1 ≤ . . . ≤ Un = G such that Ui–1 is either normal or $\mathfrak F$-normal in Ui, for i = 1,2, …, n. We call a finite group G an $fT_{\mathfrak F}$-group if every K-$\mathfrak F$-subnormal subgroup of G is in f(G). In this paper, we analyse for certain formations $\mathfrak F$ the structure of $fT_{\mathfrak F}$-groups. We pay special attention to the $\mathfrak F$-pronormal subgroups in this analysis.

scholarly journals A note on subnormal defect in finite soluble groups

1989 ◽  
Vol 39 (2) ◽  
pp. 255-258
Author(s):  
R.A. Bryce
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Soluble Groups ◽  
Sylow Subgroups ◽  
Derived Length ◽  
A Finite Group ◽  
Subnormal Subgroups

It is shown that for every positive integer n there exists a finite group of derived length n in which all Sylow subgroups are abeian and in which the defect of subnormal subgroups is at most 3.

scholarly journals The Wielandt subgroup of metacyclic p-groups

1990 ◽  
Vol 42 (3) ◽  
pp. 499-510 ◽  
Author(s):  
Elizabeth A. Ormerod
Keyword(s):  
Finite Group ◽  
Nilpotency Class ◽  
Characteristic Subgroup ◽  
Normal Series ◽  
A Finite Group ◽  
Subnormal Subgroups

The Wielandt subgroup is the intersection of the normalisers of all the subnormal subgroups of a group. For a finite group it is a non-trivial characteristic subgroup, and this makes it possible to define an ascending normal series terminating at the group. This series is called the Wielandt series and its length determines the Wielandt length of the group. In this paper the Wielandt subgroup of a metacyclic p–group is identified, and using this information it is shown that if a metacyclic p–group has Wielandt length n, its nilpotency class is n or n + 1.

scholarly journals A NOTE ON FINITE GROUPS GENERATED BY THEIR SUBNORMAL SUBGROUPS

2001 ◽  
Vol 44 (2) ◽  
pp. 417-423
Author(s):  
A. Ballester-Bolinches ◽  
L. M. Ezquerro
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Saturated Formation ◽  
Subject Classification ◽  
Primary 20F17 ◽  
A Finite Group ◽  
Fitting Formation ◽  
Subnormal Subgroups

AbstractFollowing the theory of operators created by Wielandt, we ask for what kind of formations $\mathfrak{F}$ and for what kind of subnormal subgroups $U$ and $V$ of a finite group $G$ we have that the $\mathfrak{F}$-residual of the subgroup generated by two subnormal subgroups of a group is the subgroup generated by the $\mathfrak{F}$-residuals of the subgroups.In this paper we provide an answer whenever $U$ is quasinilpotent and $\mathfrak{F}$ is either a Fitting formation or a saturated formation closed for quasinilpotent subnormal subgroups.AMS 2000 Mathematics subject classification: Primary 20F17; 20D35

Finite groups whose non-abelian self-centralizing subgroups are TI-subgroups or subnormal subgroups

2020 ◽  
pp. 2150040
Author(s):  
Yuqing Sun ◽  
Jiakuan Lu ◽  
Wei Meng
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Abelian Subgroup ◽  
Subnormal Subgroup ◽  
A Finite Group ◽  
Subnormal Subgroups

In this paper, we prove that if every non-abelian self-centralizing subgroup of a finite group [Formula: see text] is a TI-subgroup or a subnormal subgroup of [Formula: see text], then every non-abelian subgroup of [Formula: see text] must be subnormal in [Formula: see text].

scholarly journals On 𝓕-subnormal subgroups and Frattini-like subgroups of a finite group

1994 ◽  
Vol 36 (2) ◽  
pp. 241-247 ◽  
Author(s):  
A. Ballester-Bolinches ◽  
M. D. Pérez-Ramos
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Nilpotent Groups ◽  
Subnormal Subgroup ◽  
Affirmative Answer ◽  
Saturated Formation ◽  
Frattini Subgroup ◽  
A Finite Group ◽  
Subnormal Subgroups

Throughout the paper we consider only finite groups.J. C. Beidleman and H. Smith [3] have proposed the following question: “If G is a group and Ha subnormal subgroup of G containing Φ(G), the Frattini subgroup of G, such that H/Φ(G)is supersoluble, is H necessarily supersoluble? “In this paper, we give not only an affirmative answer to this question but also we see that the above result still holds if supersoluble is replaced by any saturated formation containing the class of all nilpotent groups.

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