scholarly journals A class of modules over a locally finite group III

1976 ◽  
Vol 14 (1) ◽  
pp. 95-110 ◽  
Author(s):  
B. Hartley
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Finite Subgroup ◽  
Composition Factor ◽  
Locally Finite Group ◽  
Characteristic P ◽  
Locally Finite ◽  
Soluble Groups ◽  
Group Iii ◽  
Crucial Ingredient

Let G be a locally finite group, k a field of characteristic p ≥ 0 and V a right kG-module. We say that V is an -module over kG, if each p′-subgroup H of G contains a finite subgroup F with the same fixed points as H in V. (By convention, 0′ is taken as the set of all primes.) Such modules arise as elementary abelian section of -groups, a class of locally finite groups similar in many ways to the class of finite soluble groups.The main theorem is that if V is an -module over kG with trivial Frattini submodule, and G is almost abelian, then every composition factor of V is complemented. This is a crucial ingredient in Tomkinson's theory of prefrattini subgroups in a certain subclass of . An example is given to show that the theorem breaks down for metabelian G. This leads to an example of a -group in which there are no analogues of prefrattini subgroups - the first situation where one of the standard conjugacy classes of subgroups of finite soluble groups has no decent analogue in the whole class

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

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 On Periodic Groups Saturated with Finite Frobenius Groups

2021 ◽  
Vol 35 ◽  
pp. 73-86
Author(s):  
B. E. Durakov ◽  
◽  
A. I. Sozutov ◽  
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Frobenius Group ◽  
Finite Subgroup ◽  
Frobenius Groups ◽  
Locally Finite ◽  
Periodic Groups ◽  
Group A ◽  
Quotient Groups ◽  
Combinatorial Group

A group is called weakly conjugate biprimitively finite if each its element of prime order generates a finite subgroup with any of its conjugate elements. A binary finite group is a periodic group in which any two elements generate a finite subgroup. If $\mathfrak{X}$ is some set of finite groups, then the group $G$ saturated with groups from the set $\mathfrak{X}$ if any finite subgroup of $G$ is contained in a subgroup of $G$, isomorphic to some group from $\mathfrak{X}$. A group $G = F \leftthreetimes H$ is a Frobenius group with kernel $F$ and a complement $H$ if $H \cap H^f = 1$ for all $f \in F^{\#}$ and each element from $G \setminus F$ belongs to a one conjugated to $H$ subgroup of $G$. In the paper we prove that a saturated with finite Frobenius groups periodic weakly conjugate biprimitive finite group with a nontrivial locally finite radical is a Frobenius group. A number of properties of such groups and their quotient groups by a locally finite radical are found. A similar result was obtained for binary finite groups with the indicated conditions. Examples of periodic non locally finite groups with the properties above are given, and a number of questions on combinatorial group theory are raised.

Locally finite groups and configurations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mahdi Meisami ◽  
Ali Rejali ◽  
Meisam Soleimani Malekan ◽  
Akram Yousofzadeh
Keyword(s):  
Finite Group ◽  
Positive Solution ◽  
Finite Groups ◽  
Discrete Group ◽  
Locally Finite Group ◽  
Finitely Generated ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Normalized Solution

Abstract Let 𝐺 be a discrete group. In 2001, Rosenblatt and Willis proved that 𝐺 is amenable if and only if every possible system of configuration equations admits a normalized solution. In this paper, we show independently that 𝐺 is locally finite if and only if every possible system of configuration equations admits a strictly positive solution. Also, we give a procedure to get equidecomposable subsets 𝐴 and 𝐵 of an infinite finitely generated or a locally finite group 𝐺 such that A ⊊ B A\subsetneq B , directly from a system of configuration equations not having a strictly positive solution.

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 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.

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 A class of modules over a locally finite group I

1973 ◽  
Vol 16 (4) ◽  
pp. 431-442 ◽  
Author(s):  
B. Hartley
Keyword(s):  
Finite Group ◽  
Finite Dimension ◽  
Finite Subgroup ◽  
Minimal Condition ◽  
Locally Finite Group ◽  
Locally Finite ◽  
Group I

Let G be a locally finite group, let k be a field of characteristic p ≧ 0, and let V be a (right) kG-module, not necessarily of finite dimension over k. We say that V is an Mc-modle over kG if, for each p′-subgroup H of G, the set of centrarlizers in V of subgroups of H satisfies the minimal condition under the relation of setheoretic inclusion. Here, p′ denotes the set of all peimes different from p, and in particular 0' denotes the set of all primes. It is straightforward to verify that V is an Mc-module over kG if and only if each p′-subgroup H of G contains a finite subgroup F such that CV(F) = CV(H).

scholarly journals Centralizing properties in simple locally finite groups and large finite classical groups

1990 ◽  
Vol 49 (3) ◽  
pp. 502-513 ◽  
Author(s):  
Brian Hartley
Keyword(s):  
Finite Group ◽  
Classical Group ◽  
Finite Subgroup ◽  
Locally Finite Group ◽  
Alternating Group ◽  
Local Version ◽  
Main Question ◽  
Locally Finite ◽  
Non Linear ◽  
A Finite Group

AbstractThe following question is discussed and evidence for and against it is advanced: is it true that if F is an arbitrary finite subgroup of an arbitrary non-linear simple locally finite group G, then CG(F) is infinite? The following points to an affirmative answer.Theorem A. Let F be an arbitrary finite subgroup of a non-linear simple locally finite group G. Then there exist subgroups D ◃ C ≤ G such that F centralizes C/D, F∩C ≤ D, and C/D is a direct product of finite alternating groups of unbounded orders. In particular, F centralizes an infinite section of G.Theorem A is deduced from a “local” version, namelyTheorem B. There exists an integer valued function f(n, r) with the following properties. Let H be a finite group of order at most n, and suppose that H ≤ S, where S is either an alternating group of degree at least f = f(n, r) or a finite simple classical group whose natural projective representation has degree at least f. Then there exist subgroups D ◃ C ≤ S such that (i) [H, C] ≤ D, (ii) H ∩ C ≤ D, (iii) C/D ≅ Alt(r), (iv) D = 1 if S is alternating, and D is a p-group of class at most 2 and exponent dividing p2 if S is a classical group over a field of characteristic p.The natural “local version” of our main question is however definitely false.Proposition C. Let p be a given prime. Then there exists a finite group H that can be embedded in infinitely many groups PSL(n, p) as a subgroup with trivial centralizer.

The structure of locally finite groups of finite c-dimension

2019 ◽  
Vol 18 (12) ◽  
pp. 1950223
Author(s):  
A. A. Buturlakin
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Locally Finite Group ◽  
Locally Finite ◽  
Locally Finite Groups

The [Formula: see text]-dimension of a group is the supremum of lengths of strict nested chains of centralizers. We describe the structure of locally finite groups of finite [Formula: see text]-dimension. We also prove that the [Formula: see text]-dimension of the quotient [Formula: see text] of a locally finite group [Formula: see text] by the locally soluble radical [Formula: see text] is bounded in terms of the [Formula: see text]-dimension of [Formula: see text].

Sign in / Sign up

Export Citation Format

Share Document