scholarly journals A reduction theorem for perfect locally finite minimal non-FC groups

1999 ◽  
Vol 41 (1) ◽  
pp. 81-83 ◽  
Author(s):  
FELIX LEINEN
Keyword(s):  
Conjugacy Class ◽  
Proper Subgroup ◽  
Conjugacy Classes ◽  
The Other ◽  
Reduction Theorem ◽  
Locally Finite ◽  
Open Question ◽  
Finite Conjugacy

A group G is said to be a minimal non-FC group, if G contains an infinite conjugacy class, while every proper subgroup of G merely has finite conjugacy classes. The structure of imperfect minimal non-FC groups is quite well-understood. These groups are in particular locally finite. At the other end of the spectrum, a perfect locally finite minimal non-FC group must be a p-group. And it has been an open question for quite a while now, whether such groups exist or not.

ON GROUPS WITH FINITE CONJUGACY CLASSES IN A VERBAL SUBGROUP

2017 ◽  
Vol 96 (3) ◽  
pp. 429-437 ◽  
Author(s):  
COSTANTINO DELIZIA ◽  
PAVEL SHUMYATSKY ◽  
ANTONIO TORTORA
Keyword(s):  
Conjugacy Class ◽  
Commutator Subgroup ◽  
Analogous Result ◽  
Conjugacy Classes ◽  
Verbal Subgroup ◽  
Formula Group ◽  
Finite Conjugacy

Let $w$ be a group-word. For a group $G$, let $G_{w}$ denote the set of all $w$-values in $G$ and let $w(G)$ denote the verbal subgroup of $G$ corresponding to $w$. The group $G$ is an $FC(w)$-group if the set of conjugates $x^{G_{w}}$ is finite for all $x\in G$. It is known that if $w$ is a concise word, then $G$ is an $FC(w)$-group if and only if $w(G)$ is $FC$-embedded in $G$, that is, the conjugacy class $x^{w(G)}$ is finite for all $x\in G$. There are examples showing that this is no longer true if $w$ is not concise. In the present paper, for an arbitrary word $w$, we show that if $G$ is an $FC(w)$-group, then the commutator subgroup $w(G)^{\prime }$ is $FC$-embedded in $G$. We also establish the analogous result for $BFC(w)$-groups, that is, groups in which the sets $x^{G_{w}}$ are boundedly finite.

scholarly journals GROUPS WHOSE PROPER SUBGROUPS OF INFINITE RANK HAVE FINITE CONJUGACY CLASSES

2013 ◽  
Vol 89 (1) ◽  
pp. 41-48 ◽  
Author(s):  
M. DE FALCO ◽  
F. DE GIOVANNI ◽  
C. MUSELLA ◽  
N. TRABELSI
Keyword(s):  
Conjugacy Classes ◽  
Infinite Rank ◽  
Formula Group ◽  
Open Question ◽  
Finite Conjugacy

AbstractA group $G$ is said to be an $FC$-group if each element of $G$ has only finitely many conjugates, and $G$ is minimal non$FC$ if all its proper subgroups have the property $FC$ but $G$ is not an $FC$-group. It is an open question whether there exists a group of infinite rank which is minimal non$FC$. We consider here groups of infinite rank in which all proper subgroups of infinite rank are $FC$, and prove that in most cases such groups are either $FC$-groups or minimal non$FC$.

scholarly journals Covering Theorems for FINASIGS VIII—almost all conjugacy classes in An have exponent ≤4

1978 ◽  
Vol 25 (2) ◽  
pp. 210-214 ◽  
Author(s):  
J. L. Brenner
Keyword(s):  
Conjugacy Class ◽  
Conjugacy Classes ◽  
The Other ◽  
Alternating Group ◽  
Other Hand ◽  
Covering Theorems ◽  
Image Position ◽  
Almost All

AbstractThe product of two subsets C, D of a group is defined as . The power Ce is defined inductively by C0 = {1}, Ce = CCe−1 = Ce−1C. It is known that in the alternating group An, n > 4, there is a conjugacy class C such that CC covers An. On the other hand, there is a conjugacy class D such that not only DD≠An, but even De≠An for e<[n/2]. It may be conjectured that as n ← ∞, almost all classes C satisfy C3 = An. In this article, it is shown that as n ← ∞, almost all classes C satisfy C4 = An.

scholarly journals Groups with boundedly finite conjugacy classes of commutators

2018 ◽  
Vol 69 (3) ◽  
pp. 1047-1051 ◽  
Author(s):  
Gláucia Dierings ◽  
Pavel Shumyatsky
Keyword(s):  
Conjugacy Classes ◽  
Finite Conjugacy

ON THE REGULAR GRAPH RELATED TO THE G-CONJUGACY CLASSES

2021 ◽  
pp. 1-5
Author(s):  
SH. RAHIMI ◽  
Z. AKHLAGHI
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Regular Graph ◽  
Simple Graph ◽  
Conjugacy Classes ◽  
A Finite Group

Abstract Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .

Incarnation as Awakening: Katsumi Takizawa Reading Karl Barth

Exchange ◽  
2007 ◽  
Vol 36 (2) ◽  
pp. 144-155
Author(s):  
Susanne Hennecke
Keyword(s):  
Karl Barth ◽  
The Other ◽  
Historical Jesus ◽  
Kyoto School ◽  
Dialectical Theology ◽  
Christian Theologian ◽  
The One ◽  
Open Question

AbstractThis contribution deals with the thinking of the Buddhist philosopher and Christian theologian Katsumi Takizawa (1909-1984) on incarnation. Firstly, it gives a short biographical and theological introduction to Takizawa, who was influenced not only by the "father" of the so-called dialectical theology, Karl Barth, but also by one of the famous figures of the Kyoto-school, the philosopher Kitaro Nishida.This contribution concentrates, secondly, on Takizawa's the-anthropological re-interpretation of the incarnation. It is argued that for Takizawa incarnation has to be seen as an awakening of the historical Jesus (or other historical phenomena) to what he calls the original fact: the eternal relationship between God and man.Thirdly, this contribution discusses the the-anthropological thinking of Takizawa about incarnation in five short points. Apart from the positive challenges of Takizawa's thinking especially for the theology of Karl Barth, it marks clearly the most thrilling point between Takizawa's thinking on the one side and that of scholars in Barthian theology on the other side. The open question that comes up is if incarnation really can be thought without a historical mediation or mediator, as Takizawa seems to claim.

THE CONJUGACY CLASSES OF SOME FINITE METABELIAN GROUPS AND THEIR RELATED GRAPHS

2016 ◽  
Vol 79 (1) ◽  
Author(s):  
Nor Haniza Sarmin ◽  
Ain Asyikin Ibrahim ◽  
Alia Husna Mohd Noor ◽  
Sanaa Mohamed Saleh Omer
Keyword(s):  
Graph Theory ◽  
Conjugacy Class ◽  
Dihedral Group ◽  
Clique Number ◽  
Conjugacy Classes ◽  
Quaternion Group ◽  
Metabelian Groups ◽  
Dominating Number ◽  
Graph Properties ◽  
Class Graph

In this paper, the conjugacy classes of three metabelian groups, namely the Quasi-dihedral group, Dihedral group and Quaternion group of order 16 are computed. The obtained results are then applied to graph theory, more precisely to conjugate graph and conjugacy class graph. Some graph properties such as chromatic number, clique number, dominating number and independent number are found.   

Groups with finite conjugacy classes of non-subnormal subgroups

1998 ◽  
Vol 70 (3) ◽  
pp. 169-181 ◽  
Author(s):  
S. Franciosi ◽  
F. de Giovanni ◽  
L.A. Kurdachenko
Keyword(s):  
Conjugacy Classes ◽  
Finite Conjugacy ◽  
Subnormal Subgroups

Prefrattini Subgroups and Cover-Avoidance Properties in 𝔘-Groups

1975 ◽  
Vol 27 (4) ◽  
pp. 837-851 ◽  
Author(s):  
M. J. Tomkinson
Keyword(s):  
Conjugacy Class ◽  
Soluble Group ◽  
Saturated Formation ◽  
Conjugacy Classes ◽  
Pronormal Subgroup ◽  
Finite Soluble Group ◽  
Factors Associated ◽  
Chief Factors

W. Gaschutz [5] introduced a conjugacy class of subgroups of a finite soluble group called the prefrattini subgroups. These subgroups have the property that they avoid the complemented chief factors of G and cover the rest. Subsequently, these results were generalized by Hawkes [12], Makan [14; 15] and Chambers [2]. Hawkes [12] and Makan [14] obtained conjugacy classes of subgroups which avoid certain complemented chief factors associated with a saturated formation or a Fischer class. Makan [15] and Chambers [2] showed that if W, D and V are the prefrattini subgroup, 𝔍-normalizer and a strongly pronormal subgroup associated with a Sylow basis S, then any two of W, D and V permute and the products and intersections of these subgroups have an explicit cover-avoidance property.

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