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.

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 Sylow Permutable Subnormal Subgroups of Finite Groups

2002 ◽  
Vol 251 (2) ◽  
pp. 727-738 ◽  
Author(s):  
A. Ballester-Bolinches ◽  
R. Esteban-Romero
Keyword(s):  
Finite Groups ◽  
Sylow Permutable ◽  
Subnormal Subgroups

scholarly journals Sylow permutable subnormal subgroups of finite groups II

2001 ◽  
Vol 64 (3) ◽  
pp. 479-486 ◽  
Author(s):  
A. Ballester-Bolinches ◽  
R. Esteban-Romero
Keyword(s):  
Finite Groups ◽  
Local Version ◽  
Sylow Permutable ◽  
Subnormal Subgroups

In this paper a local version of Agrawal's theorem about the structure of finite groups in which Sylow permutability is transitive is given. The result is used to obtain new characterisations of this class of finite groups.

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.

A generalized Frattini subgroup

2020 ◽  
pp. 2150026
Author(s):  
Ruslan V. Borodich
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Subnormal Subgroup ◽  
Maximal Subgroups ◽  
General Question ◽  
Frattini Subgroup ◽  
Local Formation ◽  
Glasgow Math ◽  
A Finite Group ◽  
Subnormal Subgroups

In the work of Beidleman and Smith [On Frattini-like subgroups, Glasgow Math. J. 35 (1993) 95–98], the following question was raised: “If [Formula: see text] is a subnormal subgroup of a finite group [Formula: see text] containing [Formula: see text], then whether the supersolvability of [Formula: see text] follows the supersolvability of [Formula: see text]”. This problem was considered in works of Selkin [Maximal Subgroups in the Theory of Classes of Finite Groups (Belaruskaya, Navuka, 1997)], Skiba [On the intersection of all maximal [Formula: see text]-subgroups of a finite group, Prob. Phys. Math. Tech. 3(4) (2010) 56–62], Ballester-Bolinches [On [Formula: see text]-subnormal subgroups and Frattini-like subgroups of a finite group, Glasgow Math. J. 36 (1994) 241–247] and many other authors (see monograph [Maximal Subgroups in the Theory of Classes of Finite Groups (Belaruskaya, Navuka, 1997)]). In this paper, we give the answer to the more general question: “Let [Formula: see text] be a local formation. If [Formula: see text] is a subnormal subgroup of a group [Formula: see text], then in what case [Formula: see text] will follow from [Formula: see text]”.

scholarly journals The subnormal coalescence of some classes of groups of finite rank

1973 ◽  
Vol 16 (3) ◽  
pp. 324-327 ◽  
Author(s):  
Mark Drukker ◽  
Derek J. S. Robinson ◽  
Ian Stewart
Keyword(s):  
Finite Groups ◽  
Finite Rank ◽  
Nilpotent Groups ◽  
Subnormal Subgroup ◽  
Minimal Condition ◽  
Finitely Generated ◽  
Maximal Condition ◽  
Finitely Generated Groups ◽  
Subnormal Subgroups

A class of groups forms a (subnormal) coalition class, or is (subnormally) coalescent, if wheneverHandKare subnormal -subgroups of a groupGthen their join <H, K> is also a subnormal -subgroup ofG. Among the known coalition classes are those of finite groups and polycylic groups (Wielandt [15]); groups with maximal condition for subgroups (Baer [1]); finitely generated nilpotent groups (Baer [2]); groups with maximal or minimal condition on subnormal subgroups (Robinson [8], Roseblade [11, 12]); minimax groups (Roseblade, unpublished); and any subjunctive class of finitely generated groups (Roseblade and Stonehewer [13]).

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 On perfect subnormal subgroups of finite groups

1975 ◽  
Vol 36 (2) ◽  
pp. 242-251 ◽  
Author(s):  
John S Wilson
Keyword(s):  
Finite Groups ◽  
Subnormal Subgroups
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