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.

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.

Generalized Fitting subgroup of a group of finite Morley rank

1991 ◽  
Vol 56 (4) ◽  
pp. 1391-1399 ◽  
Author(s):  
Ali Nesin
Keyword(s):  
Finite Group ◽  
Morley Rank ◽  
Fitting Subgroup ◽  
Generalized Fitting Subgroup ◽  
Finite Morley Rank ◽  
The Generalized Fitting Subgroup ◽  
A Finite Group ◽  
Subnormal Subgroups

AbstractWe define a characteristic and definable subgroup F*(G) of any group G of finite Morley rank that behaves very much like the generalized Fitting subgroup of a finite group. We also prove that semisimple subnormal subgroups of G are all definable and that there are finitely many of them.

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 RIGHT ENGEL-TYPE SUBGROUPS AND LENGTH PARAMETERS OF FINITE GROUPS

2019 ◽  
Vol 109 (3) ◽  
pp. 340-350
Author(s):  
E. I. KHUKHRO ◽  
P. SHUMYATSKY ◽  
G. TRAUSTASON
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Fitting Subgroup ◽  
Generalized Fitting Subgroup ◽  
Fitting Height ◽  
Engel Element ◽  
The Generalized Fitting Subgroup ◽  
Baer’S Theorem ◽  
A Finite Group ◽  
The Right

AbstractLet $g$ be an element of a finite group $G$ and let $R_{n}(g)$ be the subgroup generated by all the right Engel values $[g,_{n}x]$ over $x\in G$. In the case when $G$ is soluble we prove that if, for some $n$, the Fitting height of $R_{n}(g)$ is equal to $k$, then $g$ belongs to the $(k+1)$th Fitting subgroup $F_{k+1}(G)$. For nonsoluble $G$, it is proved that if, for some $n$, the generalized Fitting height of $R_{n}(g)$ is equal to $k$, then $g$ belongs to the generalized Fitting subgroup $F_{f(k,m)}^{\ast }(G)$ with $f(k,m)$ depending only on $k$ and $m$, where $|g|$ is the product of $m$ primes counting multiplicities. It is also proved that if, for some $n$, the nonsoluble length of $R_{n}(g)$ is equal to $k$, then $g$ belongs to a normal subgroup whose nonsoluble length is bounded in terms of $k$ and $m$. Earlier, similar generalizations of Baer’s theorem (which states that an Engel element of a finite group belongs to the Fitting subgroup) were obtained by the first two authors in terms of left Engel-type subgroups.

scholarly journals CENTRALIZERS OF p-SUBGROUPS IN SIMPLE LOCALLY FINITE GROUPS

2019 ◽  
Vol 62 (1) ◽  
pp. 183-186
Author(s):  
KIVANÇ ERSOY
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Fixed Points ◽  
Simple Group ◽  
Finite Groups ◽  
Abelian Subgroup ◽  
Locally Finite Group ◽  
Simple Groups ◽  
Locally Finite ◽  
Locally Finite Groups

AbstractIn Ersoy et al. [J. Algebra481 (2017), 1–11], we have proved that if G is a locally finite group with an elementary abelian p-subgroup A of order strictly greater than p2 such that CG(A) is Chernikov and for every non-identity α ∈ A the centralizer CG(α) does not involve an infinite simple group, then G is almost locally soluble. This result is a consequence of another result proved in Ersoy et al. [J. Algebra481 (2017), 1–11], namely: if G is a simple locally finite group with an elementary abelian group A of automorphisms acting on it such that the order of A is greater than p2, the centralizer CG(A) is Chernikov and for every non-identity α ∈ A the set of fixed points CG(α) does not involve an infinite simple groups then G is finite. In this paper, we improve this result about simple locally finite groups: Indeed, suppose that G is a simple locally finite group, consider a finite non-abelian subgroup P of automorphisms of exponent p such that the centralizer CG(P) is Chernikov and for every non-identity α ∈ P the set of fixed points CG(α) does not involve an infinite simple group. We prove that if Aut(G) has such a subgroup, then G ≅PSLp(k) where char k ≠ p and P has a subgroup Q of order p2 such that CG(P) = Q.

scholarly journals SCHREIER CONDITIONS ON CHIEF FACTORS AND RESIDUALS OF SOLVABLE-LIKE GROUP FORMATIONS

2008 ◽  
Vol 78 (1) ◽  
pp. 97-106
Author(s):  
GIL KAPLAN ◽  
DAN LEVY
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Solvable Groups ◽  
Fitting Subgroup ◽  
Group Extensions ◽  
Formation Of Finite Groups ◽  
Generalized Fitting Subgroup ◽  
Chief Factors ◽  
The Generalized Fitting Subgroup ◽  
Structural Characterizations

AbstractLet α be a formation of finite groups which is closed under subgroups and group extensions and which contains the formation of solvable groups. Let G be any finite group. We state and prove equivalences between conditions on chief factors of G and structural characterizations of the α-residual and theα-radical of G. We also discuss the connection of our results to the generalized Fitting subgroup of G.

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

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