scholarly journals On conditional permutability and saturated formations

2011 ◽  
Vol 54 (2) ◽  
pp. 309-319 ◽  
Author(s):  
M. Arroyo-Jordá ◽  
P. Arroyo-Jordá ◽  
M. D. Pérez-Ramos
Keyword(s):  
Saturated Formation ◽  
Finite Product ◽  
Supersoluble Groups ◽  
Soluble Groups ◽  
Permutable Subgroups ◽  
Saturated Formations

AbstractTwo subgroups A and B of a group G are said to be totally completely conditionally permutable (tcc-permutable) in G if X permutes with Yg for some g ∊ 〈X, Y〉, for all X ≤ A and Y ≤ B. We study the belonging of a finite product of tcc-permutable subgroups to a saturated formation of soluble groups containing all finite supersoluble groups.

scholarly journals Languages associated with saturated formations of groups

2015 ◽  
Vol 27 (3) ◽  
Author(s):  
Adolfo Ballester-Bolinches ◽  
Jean-Éric Pin ◽  
Xaro Soler-Escrivà
Keyword(s):  
Group Theory ◽  
Nilpotent Groups ◽  
Supersoluble Groups ◽  
Soluble Groups ◽  
Saturated Formations ◽  
Sylow Tower ◽  
General Method

AbstractIn a previous paper, the authors have shown that Eilenberg's variety theorem can be extended to more general structures, called formations. In this paper, we give a general method to describe the languages corresponding to saturated formations of groups, which are widely studied in group theory. We recover in this way a number of known results about the languages corresponding to the classes of nilpotent groups, soluble groups and supersoluble groups. Our method also applies to new examples, like the class of groups having a Sylow tower.

scholarly journals On a product of two formational tcc-subgroups

2020 ◽  
Vol 30 (2) ◽  
pp. 282-289
Author(s):  
A. Trofimuk ◽  
Keyword(s):  
Saturated Formation ◽  
Supersoluble Groups ◽  
Soluble Groups

A subgroup A of a group G is called tcc-subgroup in G, if there is a subgroup T of G such that G=AT and for any X≤A and Y≤T there exists an element u∈⟨X,Y⟩ such that XYu≤G. The notation H≤G means that H is a subgroup of a group G. In this paper we consider a group G=AB such that A and B are tcc-subgroups in G. We prove that G belongs to F, when A and B belong to F and F is a saturated formation of soluble groups such that U⊆F. Here U is the formation of all supersoluble groups.

scholarly journals ON LOCALLY DEFINED FORMATIONS OF SOLUBLE LIE AND LEIBNIZ ALGEBRAS

2012 ◽  
Vol 86 (2) ◽  
pp. 322-326 ◽  
Author(s):  
DONALD W. BARNES
Keyword(s):  
Lie Algebras ◽  
Saturated Formation ◽  
Soluble Groups ◽  
Leibniz Algebras ◽  
Saturated Formations ◽  
Nonzero Characteristic ◽  
Nilpotent Algebras

AbstractIt is well known that all saturated formations of finite soluble groups are locally defined and, except for the trivial formation, have many different local definitions. I show that for Lie and Leibniz algebras over a field of characteristic 0, the formations of all nilpotent algebras and of all soluble algebras are the only locally defined formations and the latter has many local definitions. Over a field of nonzero characteristic, a saturated formation of soluble Lie algebras has at most one local definition, but a locally defined saturated formation of soluble Leibniz algebras other than that of nilpotent algebras has more than one local definition.

FINITE GROUPS WITH GIVEN NEARLY S-QUASINORMAL SUBGROUPS

2008 ◽  
Vol 01 (03) ◽  
pp. 369-382
Author(s):  
Nataliya V. Hutsko ◽  
Vladimir O. Lukyanenko ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Cyclic Subgroup ◽  
Saturated Formation ◽  
Supersoluble Groups ◽  
Sylow Subgroups ◽  
Quasinormal Subgroup ◽  
A Finite Group

Let G be a finite group and H a subgroup of G. Then H is said to be S-quasinormal in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-quasinormal in G. Then we say that H is nearly S-quasinormal in G if G has an S-quasinormal subgroup T such that HT = G and T ∩ H ≤ HsG. Our main result here is the following theorem. Let [Formula: see text] be a saturated formation containing all supersoluble groups and G a group with a normal subgroup E such that [Formula: see text]. Suppose that every non-cyclic Sylow subgroup P of E has a subgroup D such that 1 < |D| < |P| and all subgroups H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is a non-abelian 2-group) having no supersoluble supplement in G are nearly S-quasinormal in G. Then [Formula: see text].

scholarly journals Schunck classes and projectors in a class of locally finite groups

1995 ◽  
Vol 38 (3) ◽  
pp. 511-522 ◽  
Author(s):  
M. J. Tomkinson
Keyword(s):  
Finite Groups ◽  
Maximal Subgroups ◽  
Locally Finite ◽  
Soluble Groups ◽  
Locally Finite Groups ◽  
Finite Case ◽  
Saturated Formations ◽  
Definition Of ◽  
Schunck Class

We introduce a definition of a Schunck class of periodic abelian-by-finite soluble groups using major subgroups in place of the maximal subgroups used in Finite groups. This allows us to develop the theory as in the finite case proving the existence and conjugacy of projectors. Saturated formations are examples of Schunck classes and we are also able to obtain an infinite version of Gaschütz Ω-subgroups.

ON SYLOW NORMALIZERS OF FINITE GROUPS

2013 ◽  
Vol 13 (03) ◽  
pp. 1350116 ◽  
Author(s):  
L. S. KAZARIN ◽  
A. MARTÍNEZ-PASTOR ◽  
M. D. PÉREZ-RAMOS
Keyword(s):  
Finite Groups ◽  
Property A ◽  
Soluble Groups ◽  
Sylow Subgroups ◽  
Hall Subgroups ◽  
Saturated Formations ◽  
The Universe ◽  
Do So

The paper considers the influence of Sylow normalizers, i.e. normalizers of Sylow subgroups, on the structure of finite groups. In the universe of finite soluble groups it is known that classes of groups with nilpotent Hall subgroups for given sets of primes are exactly the subgroup-closed saturated formations satisfying the following property: a group belongs to the class if and only if its Sylow normalizers do so. The paper analyzes the extension of this research to the universe of all finite groups.

scholarly journals Fitting classes of certain metanilpotent groups

1994 ◽  
Vol 36 (2) ◽  
pp. 185-195 ◽  
Author(s):  
Hermann Heinenken
Keyword(s):  
Quotient Group ◽  
Nilpotent Groups ◽  
Fitting Class ◽  
Normal Subgroups ◽  
Supersoluble Groups ◽  
Nilpotent Residual ◽  
Saturated Formations ◽  
Open Question ◽  
Fitting Classes ◽  
Cyclic Commutator

There are two families of group classes that are of particular interest for clearing up the structure of finite soluble groups: Saturated formations and Fitting classes. In both cases there is a unique conjugacy class of subgroups which are maximal as members of the respective class combined with the property of being suitably mapped by homomorphisms (in the case of saturated formations) or intersecting suitably with normal subgroups (when considering Fitting classes). While it does not seem too difficult, however, to determine the smallest saturated formation containing a given group, the same problem regarding Fitting classes does not seem answered for the dihedral group of order 6. The object of this paper is to determine the smallest Fitting class containing one of the groups described explicitly later on; all of them are qp-groups with cyclic commutator quotient group and only one minimal normal subgroup which in addition coincides with the centre. Unlike the results of McCann [7], which give a determination “up to metanilpotent groups”, the description is complete in this case. Another family of Fitting classes generated by a metanilpotent group was considered and described completely by Hawkes (see [5, Theorem 5.5 p. 476]); it was shown later by Brison [1, Proposition 8.7, Corollary 8.8], that these classes are in fact generated by one finite group. The Fitting classes considered here are not contained in the Fitting class of all nilpotent groups but every proper Fitting subclass is. They have the following additional properties: all minimal normal subgroups are contained in the centre (this follows in fact from Gaschiitz [4, Theorem 10, p. 64]) and the nilpotent residual is nilpotent of class two (answering the open question on p. 482 of Hawkes [5]), while the quotient group modulo the Fitting subgroup may be nilpotent of any class. In particular no one of these classes consists of supersoluble groups only.

On Finite Soluble Groups

1969 ◽  
Vol 9 (1-2) ◽  
pp. 250-251 ◽  
Author(s):  
J. N. Ward
Keyword(s):  
Normal Subgroup ◽  
Minimal Normal Subgroup ◽  
Saturated Formation ◽  
Fitting Subgroup ◽  
Minimal Order ◽  
Soluble Groups

Let p be a class of finite soluble groups which is closed under epimorphic images and let g be a saturated formation. Then if G is a group of minimal order belonging to p but not to g, F(G), the Fitting subgroup of G, is the unique minimal normal subgroup of G. It is to groups with this property that the following proposition is applicable.

Finite groups with given sets of 𝔉-subnormal subgroups

2019 ◽  
Vol 13 (04) ◽  
pp. 2050073 ◽  
Author(s):  
Viachaslau I. Murashka
Keyword(s):  
Finite Groups ◽  
Saturated Formation ◽  
Supersoluble Groups ◽  
Sylow Subgroups ◽  
Hereditary Saturated Formation ◽  
Subnormal Subgroups

In this paper, the classes of groups with given systems of [Formula: see text]-subnormal subgroups are studied. In particular, it is showed that if [Formula: see text] and [Formula: see text] are a saturated homomorph and a hereditary saturated formation, respectively, then the class of groups whose [Formula: see text]-subgroups are all [Formula: see text]-subnormal is a hereditary saturated formation. As corollaries, some known results about supersoluble groups, classes of groups with [Formula: see text]-subnormal cyclic primary and Sylow subgroups are obtained. Also the new characterization of the class of groups whose extreme subgroups all belong [Formula: see text], where [Formula: see text] is a hereditary saturated formation, is obtained.

scholarly journals Groups with many permutable subgroups

1990 ◽  
Vol 48 (3) ◽  
pp. 397-401 ◽  
Author(s):  
Mario Curzio ◽  
John Lennox ◽  
Akbar Rhemtulla ◽  
James Wiegold
Keyword(s):  
Free Groups ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Cyclic Subgroups ◽  
Prove Theorem ◽  
Permutable Subgroups ◽  
Infinite Set ◽  
Torsion Free

AbstractWe consider the influence on a group G of the condition that every infinite set of cyclic subgroups of G contains a pair that permute and prove (Theorem 1) that finitely generated soluble groups with this condition are centre-by-finite, and (Theorem 2) that torsion free groups satisfying the condition are abelian.

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