scholarly journals An extension of the Kegel–Wielandt theorem to locally finite groups

1996 ◽  
Vol 38 (2) ◽  
pp. 171-176
Author(s):  
Silvana Franciosi ◽  
Francesco de Giovanni ◽  
Yaroslav P. Sysak
Keyword(s):  
Finite Group ◽  
Affirmative Answer ◽  
Locally Finite Group ◽  
Linear Groups ◽  
Arbitrary Group ◽  
Famous Theorem ◽  
Locally Finite ◽  
Locally Nilpotent ◽  
Finite Products ◽  
Open Question

A famous theorem of Kegel and Wielandt states that every finite group which is the product of two nilpotent subgroups is soluble (see [1], Theorem 2.4.3). On the other hand, it is an open question whether an arbitrary group factorized by two nilpotent subgroups satisfies some solubility condition, and only a few partial results are known on this subject. In particular, Kegel [6] obtained an affirmative answer in the case of linear groups, and in the same article he also proved that every locally finite group which is the product of two locally nilpotent FC-subgroups is locally soluble. Recall that a group G is said to be an FC-group if every element of G has only finitely many conjugates. Moreover, Kazarin [5] showed that if the locally finite group G = AB is factorized by an abelian subgroup A and a locally nilpotent subgroup B, then G is locally soluble. The aim of this article is to prove the following extension of the Kegel–Wielandt theorem to locally finite products of hypercentral groups.

Groups with every subgroup ascendant-by-finite

2013 ◽  
Vol 11 (12) ◽  
Author(s):  
Sergio Camp-Mora
Keyword(s):  
Finite Group ◽  
Finite Index ◽  
Locally Finite Group ◽  
Locally Finite ◽  
Locally Nilpotent

AbstractA subgroup H of a group G is called ascendant-by-finite in G if there exists a subgroup K of H such that K is ascendant in G and the index of K in H is finite. It is proved that a locally finite group with every subgroup ascendant-by-finite is locally nilpotent-by-finite. As a consequence, it is shown that the Gruenberg radical has finite index in the whole group.

Soluble-by-periodic skew linear groups

1984 ◽  
Vol 96 (3) ◽  
pp. 379-389 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Division Ring ◽  
Matrix Algebra ◽  
Locally Finite Group ◽  
Linear Groups ◽  
Locally Finite ◽  
Full Matrix ◽  
Full Matrix Algebra ◽  
Finite Image

Let D be a division ring with central subfield F, n a positive integer and G a subgroup of GL(n, D) such that the F-subalgebra F[G] generated by G is the full matrix algebra Dn×n. If G is soluble then Snider [9] proves that G is abelian by locally finite. He also shows that this locally finite image of G can be any locally finite group. Of course not every abelian by locally finite group is soluble. This suggests that Snider's conclusion should apply to some wider class of groups.

Centralizers of subgroups in simple locally finite groups

2012 ◽  
Vol 15 (1) ◽  
Author(s):  
Kıvanç Ersoy ◽  
Mahmut Kuzucuoğlu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Finite Subgroup ◽  
Locally Finite Group ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Non Linear

AbstractHartley asked the following question: Is the centralizer of every finite subgroup in a simple non-linear locally finite group infinite? We answer a stronger version of this question for finite 𝒦-semisimple subgroups. Namely letMoreover we prove that if

scholarly journals Locally graded groups with all subgroups normal-by-finite

1996 ◽  
Vol 60 (2) ◽  
pp. 222-227 ◽  
Author(s):  
Howard Smith ◽  
James Wiegold
Keyword(s):  
Finite Group ◽  
Finite Index ◽  
Finitely Generated ◽  
Locally Finite ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Open Question ◽  
Locally Graded Groups

AbstractIn a paper published in this journal [1], J. T. Buckely, J. C. Lennox, B. H. Neumann and the authors considered the class of CF-groups, that G such that |H: CoreG (H)| is finite for all subgroups H. It is shown that locally finite CF-groups are abelian-by-finite and BCF, that is, there is an integer n such that |H: CoreG(H)| ≤ n for all subgroups H. The present paper studies these properties in the class of locally graded groups, the main result being that locally graded BCF-groups are abelian-by-finite. Whether locally graded CF-groups are BFC remains an open question. In this direction, the following problems is posed. Does there exist a finitely generated infinite periodic residually finite group in which all subgroups are finite or of finite index? Such groups are locally graded and CF but not BCF.

A NOTE ON FIXED RINGS

2005 ◽  
Vol 04 (02) ◽  
pp. 165-171
Author(s):  
HAKIMA MOUANIS
Keyword(s):  
Finite Group ◽  
Quotient Ring ◽  
Locally Finite Group ◽  
Prime Ideals ◽  
Locally Finite ◽  
Group Of Automorphisms ◽  
Fixed Ring ◽  
Minimal Prime

Let R be a ring and G a group of automorphisms of R. In this paper we investigate the transfer of the Von Neumman regularity from the total quotient ring of R to the total quotient ring of its fixed ring. We prove also that, for a locally finite group, Min(RG) inherits the compactness from Min(R), where Min(R) denotes the set of all minimal prime ideals of R. Finally, We explore some conditions under which we can transfer the PF-property (respectively, pseudo PF-property) from a ring to its fixed subring.

Local systems and Sylow subgroups in locally finite groups. II

1974 ◽  
Vol 75 (1) ◽  
pp. 1-22 ◽  
Author(s):  
A. Rae
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Local System ◽  
The Other ◽  
Locally Finite Group ◽  
Local Systems ◽  
Locally Finite ◽  
Sylow Subgroups ◽  
Locally Finite Groups ◽  
Other Hand

1.1. Introduction. In this paper, we continue with the theme of (1): the relationships holding between the Sπ (i.e. maximal π) subgroups of a locally finite group and the various local systems of that group. In (1), we were mainly concerned with ‘good’ Sπ subgroups – those which reduce into some local system (and are said to be good with respect to that system). Here, on the other hand, we are concerned with a very much more special sort of Sπ subgroup.

On locally nilpotent groups

1956 ◽  
Vol 52 (1) ◽  
pp. 5-11 ◽  
Author(s):  
D. H. McLain ◽  
P. Hall
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Direct Product ◽  
Finite Subset ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Locally Finite ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent ◽  
A Finite Group

1. If P is any property of groups, then we say that a group G is ‘locally P’ if every finitely generated subgroup of G satisfies P. In this paper we shall be chiefly concerned with the case when P is the property of being nilpotent, and will examine some properties of nilpotent groups which also hold for locally nilpotent groups. Examples of locally nilpotent groups are the locally finite p-groups (groups such that every finite subset is contained in a finite group of order a power of the prime p); indeed, every periodic locally nilpotent group is the direct product of locally finite p-groups.

scholarly journals Irreducible locally nilpotent finitary skew linear groups

1995 ◽  
Vol 38 (1) ◽  
pp. 63-76 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Linear Group ◽  
Division Ring ◽  
Linear Groups ◽  
Locally Finite ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finitary Linear Group ◽  
Locally Nilpotent ◽  
Finitary Linear ◽  
Finite Dimensional Case

Let V be a left vector space over the arbitrary division ring D and G a locally nilpotent group of finitary automorphisms of V (automorphisms g of V such that dimDV(g-1)<∞) such that V is irreducible as D-G bimodule. If V is infinite dimensional we show that such groups are very rare, much rarer than in the finite-dimensional case. For example we show that if dimDV is infinite then dimDV = |G| = ℵ0 and G is a locally finite q-group for some prime q ≠ char D. Moreover G is isomorphic to a finitary linear group over a field. Examples show that infinite-dimensional such groups G do exist. Note also that there exist examples of finite-dimensional such groups G that are not isomorphic to any finitary linear group over a field. Generally the finite-dimensional examples are more varied.

scholarly journals COMMUTATIVITY DEGREE OF A CLASS OF FINITE GROUPS AND CONSEQUENCES

2013 ◽  
Vol 88 (3) ◽  
pp. 448-452 ◽  
Author(s):  
RAJAT KANTI NATH
Keyword(s):  
Finite Group ◽  
Semidirect Product ◽  
Finite Groups ◽  
Abelian Groups ◽  
Affirmative Answer ◽  
Finite Abelian Groups ◽  
A Finite Group ◽  
Open Question

AbstractThe commutativity degree of a finite group is the probability that two randomly chosen group elements commute. The object of this paper is to compute the commutativity degree of a class of finite groups obtained by semidirect product of two finite abelian groups. As a byproduct of our result, we provide an affirmative answer to an open question posed by Lescot.

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