scholarly journals Finite groups in which normality, permutability or Sylow permutability is transitive

2014 ◽  
Vol 22 (3) ◽  
pp. 137-146
Author(s):  
Izabela Agata Malinowska
Keyword(s):  
Finite Groups ◽  
Subnormal Subgroup ◽  
Soluble Groups ◽  
Sylow Permutable

AbstractY. Li gave a characterization of the class of finite soluble groups in which every subnormal subgroup is normal by means of NE-subgroups: a subgroup H of a group G is called an NE-subgroup of G if NG(H) ∩ HG = H. We obtain a new characterization of these groups related to the local Wielandt subgroup. We also give characterizations of the classes of finite soluble groups in which every subnormal subgroup is permutable or Sylow permutable in terms of NE-subgroups.

scholarly journals The subnormal structure of some classes of coluble groups

1972 ◽  
Vol 13 (3) ◽  
pp. 365-377 ◽  
Author(s):  
D. McDougall
Keyword(s):  
Finite Groups ◽  
Soluble Group ◽  
Subnormal Subgroup ◽  
Negative Integer ◽  
Soluble Groups ◽  
Derived Length ◽  
Subnormal Structure ◽  
A Chain

Finite groups in which normality is transitive have been studied by Best and Taussky, [1], Gaschütz, [3], and Zacher [16]. Infinite soluble groups in which normality is transitive have been studied by Robinson in [9]. A subgroup H of a group G is subnormal in G if H can be connected to G by a chain of r subgroups, in which each is normal in its successor, where r is a non-negative integer. The least such r is called the subnormal index of H in G (or the defect of H in G). Then groups in which normality is transitive are precisely those in which every subnormal subgroup has subnormal index at most one. Thus the structure of soluble groups in which every subnormal subgroup has subnormal index at most n (such a group is said to have bounded subnormal indices) has been dealt with by Robinson in [9] for the case where n is one. However Theorem D of [12] states that a soluble group of derived length n can be embedded in a soluble group in which the subnormal indices are at most n. Therefore we must impose further conditions on the groups if we hope to obtain any worthwhile results for the above problem with n greater than one.

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.

Finite Groups Whose Cyclic Subnormal Subgroups Are Permutable

2005 ◽  
Vol 12 (01) ◽  
pp. 171-180 ◽  
Author(s):  
Derek J. S. Robinson
Keyword(s):  
Finite Groups ◽  
Subnormal Subgroup ◽  
Sylow Permutable ◽  
Subnormal Subgroups

Finite groups in which each cyclic subnormal subgroup is Sylow-permutable, permutable or normal are investigated. Characterizations are found in both the soluble and insoluble cases.

Finite groups with given systems of σ-semipermutable subgroups

2018 ◽  
Vol 17 (02) ◽  
pp. 1850031 ◽  
Author(s):  
Bin Hu ◽  
Jianhong Huang ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Chief Factor ◽  
Transitive Relation ◽  
Soluble Groups ◽  
A Finite Group

Let [Formula: see text] be a partition of the set of all primes [Formula: see text] and [Formula: see text] a finite group. [Formula: see text] is said to be [Formula: see text]-soluble if every chief factor [Formula: see text] of [Formula: see text] is a [Formula: see text]-group for some [Formula: see text]. A set [Formula: see text] of subgroups of [Formula: see text] is said to be a complete Hall [Formula: see text]-set of [Formula: see text] if every member [Formula: see text] of [Formula: see text] is a Hall [Formula: see text]-subgroup of [Formula: see text] for some [Formula: see text] and [Formula: see text] contains exactly one Hall [Formula: see text]-subgroup of [Formula: see text] for every [Formula: see text] such that [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-quasinormal or [Formula: see text]-permutable in [Formula: see text] if [Formula: see text] has a complete Hall [Formula: see text]-set [Formula: see text] such that [Formula: see text] for all [Formula: see text] and all [Formula: see text]. We obtain a new characterization of finite [Formula: see text]-soluble groups [Formula: see text] in which [Formula: see text]-permutability is a transitive relation in [Formula: see text].

Finite Groups with Some Subgroups of Sylow Subgroups s∗-Semipermutable

2021 ◽  
Vol 58 (2) ◽  
pp. 147-156
Author(s):  
Qingjun Kong ◽  
Xiuyun Guo
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Subnormal Subgroup ◽  
Sylow Subgroups ◽  
A Finite Group

We introduce a new subgroup embedding property in a finite group called s∗-semipermutability. Suppose that G is a finite group and H is a subgroup of G. H is said to be s∗-semipermutable in G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K is s-semipermutable in G. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying 1 < |D| < |P | and study the structure of G under the assumption that every subgroup H of P with |H | = |D| is s∗-semipermutable in G. Some recent results are generalized and unified.

THE INFLUENCE OF Φ-S-SUPPLEMENTED SUBGROUPS ON THE STRUCTURE OF FINITE GROUPS

2012 ◽  
Vol 11 (04) ◽  
pp. 1250064
Author(s):  
CHANGWEN LI
Keyword(s):  
Finite Groups ◽  
Subnormal Subgroup ◽  
Frattini Subgroup

A subgroup H of a group G is called Φ-s-supplemented in G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K ≤ Φ (H), where Φ(H) is the Frattini subgroup of H. We investigate the influence of Φ-s-supplemented subgroups on the p-nilpotency, p-supersolvability and supersolvability of finite groups.

scholarly journals Quasinormal Fitting classes of finite groups

2019 ◽  
pp. 18-26
Author(s):  
Martsinkevich Anna V.
Keyword(s):  
Natural Number ◽  
Wreath Product ◽  
Cyclic Group ◽  
Finite Groups ◽  
Fitting Class ◽  
Soluble Groups ◽  
Normal Fitting Class ◽  
Fitting Classes ◽  
Local Fitting ◽  
Class F

Let P be the set of all primes, Zn a cyclic group of order n and X wr Zn the regular wreath product of the group X with Zn. A Fitting class F is said to be X-quasinormal (or quasinormal in a class of groups X ) if F ⊆ X, p is a prime, groups G ∈ F and G wr Zp ∈ X, then there exists a natural number m such that G m wr Zp ∈ F. If  X is the class of all soluble groups, then F is normal Fitting class. In this paper we generalize the well-known theorem of Blessenohl and Gaschütz in the theory of normal Fitting classes. It is proved, that the intersection of any set of nontrivial X-quasinormal Fitting classes is a nontrivial X-quasinormal Fitting class. In particular, there exists the smallest nontrivial X-quasinormal Fitting class. We confirm a generalized version of the Lockett conjecture (in particular, the Lockett conjecture) about the structure of a Fitting class for the case of X-quasinormal classes, where X is a local Fitting class of partially soluble groups.

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