scholarly journals Permutable Subgroups in GLn(D) and Applications to Locally Finite Group Algebras

2021 ◽  
Author(s):  
Le Qui Danh ◽  
Mai Hoang Bien ◽  
Bui Xuan Hai
Keyword(s):  
Finite Group ◽  
Locally Finite Group ◽  
Group Algebras ◽  
Locally Finite ◽  
Permutable Subgroups

Injective modules for group algebras of locally finite groups

1978 ◽  
Vol 84 (2) ◽  
pp. 247-262 ◽  
Author(s):  
I. M. Musson
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Countable Group ◽  
Locally Finite Group ◽  
Group Algebras ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Injective Modules

Two recent results relate the existence of injective modules for group algebras which are ‘small’ in some sense to the structure of the group.(1) The trivial kG-module is injective if and only if G is a locally finite group with no elements of order p = char k (9).(2) If (G) is a countable group, then every irreducible kG-module is injective if and only if G is a locally finite p′ group which is abelian-by-finite (9) and (11)

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.

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.

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

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.

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.

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.

scholarly journals Group algebras whose p-elements form a subgroup

2016 ◽  
Vol 16 (09) ◽  
pp. 1750170
Author(s):  
M. Ramezan-Nassab
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Group Algebra ◽  
Polynomial Identity ◽  
Matrix Polynomial ◽  
Unit Group ◽  
Group Algebras ◽  
Locally Finite ◽  
P Elements

Let [Formula: see text] be a group, [Formula: see text] a field of characteristic [Formula: see text], and [Formula: see text] the unit group of the group algebra [Formula: see text]. In this paper, among other results, we show that if either (1) [Formula: see text] satisfies a non-matrix polynomial identity, or (2) [Formula: see text] is locally finite, [Formula: see text] is infinite and [Formula: see text] is an Engel-by-finite group, then the [Formula: see text]-elements of [Formula: see text] form a (normal) subgroup [Formula: see text] and [Formula: see text] is abelian (here, of course, [Formula: see text] if [Formula: see text]).

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.

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