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 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.

scholarly journals Nilpotent-like fitting formations of finite soluble groups

2000 ◽  
Vol 62 (3) ◽  
pp. 427-433 ◽  
Author(s):  
A. Ballester-Bolinches ◽  
M. D. Pérez-Ramos ◽  
A. Martínez- Pastor
Keyword(s):  
Nilpotent Groups ◽  
Subnormal Subgroup ◽  
60Th Birthday ◽  
Soluble Groups ◽  
Saturated Formations

Dedicated to Professor K. Doerk on his 60th Birthday.In this paper the subnormal subgroup closed saturated formations of finite soluble groups containing nilpotent groups are fully characterised by means of extensions of well-known properties enjoyed by the formation of all nilpotent groups.

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 Supersoluble groups of automorphisms of compact Riemann surfaces

1989 ◽  
Vol 31 (3) ◽  
pp. 321-327 ◽  
Author(s):  
Grzegorz Gromadzki ◽  
Colin MacLachlan
Keyword(s):  
Finite Groups ◽  
Riemann Surfaces ◽  
Nilpotent Groups ◽  
Metabelian Groups ◽  
Supersoluble Groups ◽  
Soluble Groups ◽  
Cyclic Groups ◽  
Groups Of Automorphisms ◽  
Compact Riemann Surfaces ◽  
Infinite Sequences

Given an integer g ≥ 2 and a class of finite groups let N(g, ) denote the order of the largest group in that a compact Riemann surface of genus g admits as a group of automorphisms. For the classes of all finite groups, cyclic groups, abelian groups, nilpotent groups, p-groups (given p), soluble groups and finally for metabelian groups, an upper bound for N(g, ) as well as infinite sequences for g for which this bound is attained were found in [5, 6, 7, 8, 13], [4], [10], [15], [16], [1], [2] respectively. This paper deals with that problem for the class of finite supersoluble groups i.e. groups with an invariant series all of whose factors are cyclic. In addition, it goes further by describing exactly those values of g for which the bound is attained. More precisely we prove:

A family of Fitting classes of supersoluble groups

1995 ◽  
Vol 118 (1) ◽  
pp. 49-57 ◽  
Author(s):  
Martin Menth
Keyword(s):  
Soluble Group ◽  
Nilpotent Groups ◽  
Fitting Class ◽  
Finite Soluble Group ◽  
Supersoluble Groups ◽  
Fitting Classes ◽  
General Method ◽  
Subnormal Subgroups

A class of groups that is closed with respect to subnormal subgroups and normal products is called a Fitting class. Given a finite soluble group G, one may ask for the Fitting class (G) generated by G, that is the intersection of all Fitting classes containing G. For simple or nilpotent groups G it is easy to compute (G), but in other cases the determination of (G) seems to be surprisingly difficult, and there is no general method of solving this problem. In recent years there has been a lot of work in this area, see for instance Bryce and Cossey[l], [2], Hawkes[6] (or [5], IX. 9. Var. II), Heineken[7] and McCann[10].

scholarly journals Locally soluble groups with min-n.

1974 ◽  
Vol 17 (3) ◽  
pp. 305-318 ◽  
Author(s):  
H. Heineken ◽  
J. S. Wilson
Keyword(s):  
Soluble Group ◽  
Free Groups ◽  
Torsion Group ◽  
Minimal Condition ◽  
Normal Subgroups ◽  
Soluble Groups ◽  
Isomorphism Classes ◽  
Torsion Groups ◽  
Torsion Free ◽  
General Method

It was shown by Baer in [1] that every soluble group satisfying Min-n, the minimal condition for normal subgroups, is a torsion group. Examples of non-soluble locally soluble groups satisfying Min-n have been known for some time (see McLain [2]), and these examples too are periodic. This raises the question whether all locally soluble groups with Min-n are torsion groups. We prove here that this is not the case, by establishing the existence of non-trivial locally soluble torsion-free groups satisfying Min-n. Rather than exhibiting one such group G, we give a general method for constructing examples; the reader will then be able to see that a variety of additional conditions may be imposed on G. It will follow, for instance, that G may be a Hopf group whose normal subgroups are linearly ordered by inclusion and are all complemented in G; further, that the countable groups G with these properties fall into exactly isomorphism classes. Again, there are exactly isomorphism classes of countable groups G which have hypercentral nonnilpotent Hirsch-Plotkin radical, and which at the same time are isomorphic to all their non-trivial homomorphic images.

scholarly journals Normal -Fitting classes

1992 ◽  
Vol 35 (2) ◽  
pp. 201-212
Author(s):  
J. C. Beidleman ◽  
M. J. Tomkinson
Keyword(s):  
Nilpotent Groups ◽  
Fitting Class ◽  
Soluble Groups ◽  
Locally Nilpotent ◽  
Normal Fitting Class ◽  
Fitting Classes

The authors together with M. J. Karbe [Ill. J. Math. 33 (1989) 333–359] have considered Fitting classes of -groups and, under some rather strong restrictions, obtained an existence and conjugacy theorem for -injectors. Results of Menegazzo and Newell show that these restrictions are, in fact, necessary.The Fitting class is normal if, for each is the unique -injector of G. is abelian normal if, for each. For finite soluble groups these two concepts coincide but the class of Černikov-by-nilpotent -groups is an example of a nonabelian normal Fitting class of -groups. In all known examples in which -injectors exist is closely associated with some normal Fitting class (the Černikov-by-nilpotent groups arise from studying the locally nilpotent injectors).Here we investigate normal Fitting classes further, paying particular attention to the distinctions between abelian and nonabelian normal Fitting classes. Products and intersections with (abelian) normal Fitting classes lead to further examples of Fitting classes satisfying the conditions of the existence and conjugacy theorem.

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.

Sign in / Sign up

Export Citation Format

Share Document