Splitting over nilpotent and hypercentral residuals

1975 ◽  
Vol 78 (2) ◽  
pp. 215-226 ◽  
Author(s):  
B. Hartley ◽  
M. J. Tomkinson
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Formation Theory ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Nilpotent Residual ◽  
A Finite Group

It is a well known theorem of Gaschütz (4) and Schenkman (12) that if G is a finite group whose nilpotent residual A is Abelian, then G splits over A and the complements to A in G are conjugate. Following Robinson (10) we describe this situation by saying that G splits conjugately over A. A number of generalizations of this result have since been obtained, some of them being in the context of the formation theory of finite or locally finite groups (see, for example, (1), (3)) and others, for example, the recent and far-reaching results of Robinson (10, 11) being concerned with groups which are not necessarily periodic. Our results here are of the latter type.

Finite groups with two supersoluble subgroups

2019 ◽  
Vol 22 (2) ◽  
pp. 297-312 ◽  
Author(s):  
Victor S. Monakhov ◽  
Alexander A. Trofimuk
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Maximal Subgroups ◽  
Sylow Subgroups ◽  
Derived Subgroup ◽  
Nilpotent Residual ◽  
A Finite Group

AbstractLetGbe a finite group. In this paper we obtain some sufficient conditions for the supersolubility ofGwith two supersoluble non-conjugate subgroupsHandKof prime index, not necessarily distinct. It is established that the supersoluble residual of such a group coincides with the nilpotent residual of the derived subgroup. We prove thatGis supersoluble in the following cases: one of the subgroupsHorKis nilpotent; the derived subgroup{G^{\prime}}ofGis nilpotent;{|G:H|=q>r=|G:K|}andHis normal inG. Also the supersolubility ofGwith two non-conjugate maximal subgroupsMandVis obtained in the following cases: all Sylow subgroups ofMand ofVare seminormal inG; all maximal subgroups ofMand ofVare seminormal inG.

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.

Formation theory and groups of automorphisms of -groups

1977 ◽  
Vol 76 (4) ◽  
pp. 255-265 ◽  
Author(s):  
M. J. Tomkinson
Keyword(s):  
Finite Groups ◽  
Composition Series ◽  
Formation Theory ◽  
Locally Finite ◽  
Soluble Groups ◽  
Locally Finite Groups ◽  
Group Of Automorphisms ◽  
Infinite Case ◽  
Groups Of Automorphisms ◽  
Automorphisms Of Groups

SynopsisFurther results from the theory of finite soluble groups are extended to the class of locally finite groups with a satisfactory Sylow structure. Let be a saturated U-formation and A a -group of automorphisms of the -group G. A is said to act -centrally on G if G has an A-composition series (Λσ/Vσ; σ ∈ ∑) such that A induces an f(p)-group of automorphisms in each p-factor Λσ/Vσ. We show that in this situation A is an -group, thus generalising the result of Schmid [8]. Associated results of Schmid and of Baer are also extended to the infinite case.

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.

scholarly journals ON GROUPS WITH A FINITE NUMBER OF NORMALISERS

2012 ◽  
Vol 86 (3) ◽  
pp. 416-423 ◽  
Author(s):  
MOHAMMAD ZARRIN
Keyword(s):  
Finite Group ◽  
Finite Number ◽  
Finite Groups ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
As Solubility

AbstractGroups having exactly one normaliser are well known. They are the Dedekind groups. All finite groups having exactly two normalisers were classified by Pérez-Ramos [‘Groups with two normalizers’, Arch. Math.50 (1988), 199–203], and Camp-Mora [‘Locally finite groups with two normalizers’, Comm. Algebra28 (2000), 5475–5480] generalised that result to locally finite groups. Then Tota [‘Groups with a finite number of normalizer subgroups’, Comm. Algebra32 (2004), 4667–4674] investigated properties (such as solubility) of arbitrary groups with two, three and four normalisers. In this paper we prove that every finite group with at most 20 normalisers is soluble. Also we characterise all nonabelian simple (not necessarily finite) groups with at most 57 normalisers.

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