scholarly journals Fitting classes of CC-groups

1988 ◽  
Vol 31 (3) ◽  
pp. 475-479 ◽  
Author(s):  
Martyn R. Dixon
Keyword(s):  
Recent Work ◽  
Present Note ◽  
Standard Group ◽  
Infinite Groups ◽  
Soluble Groups ◽  
Analogous Manner ◽  
Scant Attention ◽  
Černikov Group ◽  
Fitting Classes

The theory of Fitting classes is, by now, a well established part of the theory of finite soluble groups. In contrast, Fitting classes have received rather scant attention in infinite groups, although some recent work of Beidleman and Karbe [2] and Beidleman, Karbe and Tomkinson [3] suggest that one can obtain results in this direction. The paper [2], cited above, in fact generalizes earlier work of Tomkinson [9] to the class of locally soluble FC-groups. The present paper is concerned with the theory of Fitting classes in a class of groups somewhat similar to the class of FC-groups, namely the class of CC-groups, introduced by Polovickiǐ in [6]. A group G is a CC-group if G/CG(xG) is a Černikov group for all x ∈ G where, as in the rest of this paper, we use the standard group theoretic notation of [7]. Recently, Alcázar and Otal [1] have shown how to generalize results of B. H. Neumann [5] to the class of CC-groups. The main purpose of the present note is to illustrate further how one can handle CC-groups, in an analogous manner to FC-groups, by using techniques similar to those used in [1] and [4].

scholarly journals A family of dominant Fitting classes of finite soluble groups

1998 ◽  
Vol 64 (1) ◽  
pp. 33-43 ◽  
Author(s):  
A. Ballester-Bolinches ◽  
A. Martínez-Pastor ◽  
M. D. Pérez-Ramos
Keyword(s):  
Large Family ◽  
Soluble Groups ◽  
Fitting Classes

AbstractIn this paper a large family of dominant Fitting classes of finite soluble groups and the description of the corresponding injectors are obtained. Classical constructions of nilpotent and Lockett injectors as well as p-nilpotent injectors arise as particular cases.

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.

scholarly journals Maximal Fitting classes of finite soluble groups

1974 ◽  
Vol 10 (2) ◽  
pp. 169-175 ◽  
Author(s):  
R.A. Bryce ◽  
John Cossey
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Fitting Class ◽  
Soluble Groups ◽  
Necessary And Sufficient ◽  
Fitting Classes

From recent results of Lausch, it is easy to establish necessary and sufficient conditions for a Fitting class to be maximal in the class of all finite soluble groups. We use Lausch's methods to show that there are normal Fitting classes not contained in any Fitting class maximal in the class of all finite soluble groups. We also find conditions on Fitting classes and for to be maximal in .

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 Conjugate purity and infinite groups

1979 ◽  
Vol 28 (1) ◽  
pp. 9-14
Author(s):  
Bola O. Balogun
Keyword(s):  
Finite Group ◽  
Subject Classification ◽  
Infinite Groups ◽  
Soluble Groups ◽  
Infinite Case ◽  
A Finite Group

AbstractIn Balogun (1974), we proved that a finite group in which every subgroup is conjugately pure is necessarily Abelian and we left open the infinite case. In this paper we settle this problem positively for soluble, locally soluble groups and certain classes of groups which include the FC-groups. In the last section of this paper we characterize groups which are conjugately pure in every containing group.Subject classification (Amer. Math. Soc. (MOS) 1970): 20 E 99.

scholarly journals A criterion for the Hall-closure of Fitting classes

1981 ◽  
Vol 23 (3) ◽  
pp. 361-365 ◽  
Author(s):  
Owen J. Brison
Keyword(s):  
Fitting Class ◽  
Soluble Groups ◽  
Normal Fitting Class ◽  
Fitting Classes

In a recent paper, Cusack has given a criterion, in terms of the Fitting class “join” operation, for a normal Fitting class to be closed under the taking of Hall π-subgroups. Here we show that Cusack's result can be slightly modified so as to give a criterion for any Fitting class of finite soluble groups to be closed under taking Hall π-subgroups.

scholarly journals The lattice of Fitting classes which are right extensible by soluble groups

2005 ◽  
Vol 49 ◽  
pp. 351-362
Author(s):  
M. J. Iranzo ◽  
J. P. Lafuente ◽  
F. Pérez-Monasor
Keyword(s):  
Soluble Groups ◽  
Fitting Classes

scholarly journals Hall-closure and products of Fitting classes

1982 ◽  
Vol 32 (2) ◽  
pp. 145-164 ◽  
Author(s):  
Owen J. Brison
Keyword(s):  
Fitting Class ◽  
Soluble Groups ◽  
Normal Fitting Class ◽  
Group Construction ◽  
Case Part ◽  
Fitting Classes

AbstractThe Fitting class (of finite, soluble, groups), , is said to be Hall π-closed (where π is a set of primes) if whenever G is a group in and H is a Hall π-subgroup of G, then H belongs to . In this paper, we study the Hall π-closure of products of Fitting classes. Our main result is a characterisation of the Hall π-closedFitting classes of the form (where denotes the so-called smallest normal Fitting class), subject to a restriction connecting π with the characteristic of . We also characterise those Fitting classes (respectively, ) such that (respectively, ) is Hall π-closed for all Fitting classes . In each case, part of the proof uses a concrete group construction. As a bonus, one of these construction also yields a “cancellation result” for certain products of Fitting classes.

Groups with Finitely Many Homomorphic Images of Finite Rank

2016 ◽  
Vol 23 (02) ◽  
pp. 181-187
Author(s):  
Francesco de Giovanni ◽  
Alessio Russo
Keyword(s):  
Large Class ◽  
Finite Rank ◽  
Minimal Condition ◽  
Soluble Groups ◽  
Normal Series ◽  
Černikov Group

A group is called a Černikov group if it is abelian-by-finite and satisfies the minimal condition on subgroups. A new characterization of Černikov groups is given here, by proving that in a suitable large class of generalised soluble groups they coincide with the groups having only finitely many homomorphic images of finite rank (up to isomorphisms) and admitting an ascending normal series whose factors have finite rank.

Some Sylow subgroups

1961 ◽  
Vol 260 (1302) ◽  
pp. 304-316 ◽  
Keyword(s):  
Finite Groups ◽  
Infinite Groups ◽  
Soluble Groups ◽  
Sylow Subgroups ◽  
Uncountable Group

Most of the well-known theorems of Sylow for finite groups and of P. Hall for finite soluble groups have been extended to certain restricted classes of infinite groups. To show the limitations of such generalizations, examples are here constructed of infinite groups subject to stringent but natural restrictions, groups in which certain Sylow or Hall theorems fail. All the groups are metabelian and of exponent 6. There are countable such groups in which a Sylow 2-subgroup has a complement but no Sylow 3-complement; or again no complement at all. There are countable such groups with continuously many mutually non-isomorphic Sylow 3-subgroups. There are groups, necessarily of uncountable order, with Sylow 2-subgroups of different orders. The most elaborate example is of an uncountable group in which all Sylow 2-subgroups and 3-subgroups are countable, and none is complemented.

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