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.

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.

scholarly journals Two remarks on amalgams of locally finite groups

1987 ◽  
Vol 36 (3) ◽  
pp. 461-468 ◽  
Author(s):  
Berthold J. Maier
Keyword(s):  
Finite Group ◽  
Finite Index ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Locally Finite Group ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
The Common ◽  
Necessary And Sufficient

We construct non amalgamation bases in the class of locally finite groups, and we present necessary and sufficient conditions for the embeddability of an amalgam into a locally finite group in the case that the common subgroup has finite index in both constituents.

scholarly journals ON THE CENTRALIZER OF AN ELEMENT OF ORDER FOUR IN A LOCALLY FINITE GROUP

2007 ◽  
Vol 49 (2) ◽  
pp. 411-415 ◽  
Author(s):  
PAVEL SHUMYATSKY
Keyword(s):  
Finite Group ◽  
Finite Index ◽  
Locally Finite Group ◽  
Locally Finite ◽  
Soluble Subgroup ◽  
Order Four

AbstractWe prove that if G is a locally finite group admitting an automorphism φ of order four such that CG(φ) is Chernikov, then G has a soluble subgroup of finite index.

scholarly journals LOCALLY FINITE GROUPS WHOSE SUBGROUPS HAVE FINITE NORMAL OSCILLATION

2013 ◽  
Vol 89 (3) ◽  
pp. 479-487 ◽  
Author(s):  
F. DE GIOVANNI ◽  
M. MARTUSCIELLO ◽  
C. RAINONE
Keyword(s):  
Finite Group ◽  
Finite Index ◽  
Finite Groups ◽  
Cardinal Number ◽  
Nilpotent Subgroup ◽  
Locally Finite Group ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Normal Oscillation

AbstractIf $X$ is a subgroup of a group $G$, the cardinal number $\min \{ \vert X: X_{G}\vert , \vert {X}^{G} : X\vert \} $ is called the normal oscillation of $X$ in $G$. It is proved that if all subgroups of a locally finite group $G$ have finite normal oscillation, then $G$ contains a nilpotent subgroup of finite index.

scholarly journals The structure of groups whose subgroups are permutable-by-finite

2006 ◽  
Vol 81 (1) ◽  
pp. 35-48 ◽  
Author(s):  
M. De Falco ◽  
F. De Giovanni ◽  
C. Musella ◽  
Y. P. Sysak
Keyword(s):  
Finite Group ◽  
Finite Index ◽  
Abelian Subgroup ◽  
Locally Finite Group ◽  
Normal Subgroups ◽  
Locally Finite ◽  
Permutable Subgroups

AbstractA subgroup H of a group G is said to be permutable if HX = XH for each subgroup X of G, and the group G is called quasihamiltonian if all its subgroups are permutable. We shall say that G is a Q F-group if every subgroup H of G contains a subgroup K of finite index which is permutable in G. It is proved that every locally finite Q F-group contains a quasihamiltonian subgroup of finite index. In the proof of this result we use a theorem by Buckley, Lennox, Neumann, Smith and Wiegold concerning the corresponding problem when permutable subgroups are replaced by normal subgroups: if G is a locally finite group such that H/HG is finite for every subgroup H, then G contains an abelian subgroup of finite index.

scholarly journals Groups with bounded centralizer chains and the Borovik–Khukhro conjecture

2018 ◽  
Vol 21 (6) ◽  
pp. 1095-1110
Author(s):  
Alexander A. Buturlakin ◽  
Danila O. Revin ◽  
Andrey V. Vasil’ev
Keyword(s):  
Finite Group ◽  
Finite Index ◽  
Abelian Subgroup ◽  
Inverse Image ◽  
Locally Finite Group ◽  
Fitting Subgroup ◽  
Locally Finite ◽  
Generalized Fitting Subgroup ◽  
The Generalized Fitting Subgroup ◽  
Weak Analogue

Abstract Let G be a locally finite group and let {F(G)} be the Hirsch–Plotkin radical of G. Let S denote the full inverse image of the generalized Fitting subgroup of {G/F(G)} in G. Assume that there is a number k such that the length of every nested chain of centralizers in G does not exceed k. The Borovik–Khukhro conjecture states, in particular, that under this assumption, the quotient {G/S} contains an abelian subgroup of finite index bounded in terms of k. We disprove this statement and prove a weak analogue of it.

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.

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