Locally graded groups with complemented infinite nonabelian subgroups

1992 ◽  
Vol 43 (7-8) ◽  
pp. 1031-1033 ◽  
Author(s):  
B. I. Mishchenko
Keyword(s):  
Locally Graded Groups

scholarly journals Group Laws Implying Virtual Nilpotence

2003 ◽  
Vol 74 (3) ◽  
pp. 295-312 ◽  
Author(s):  
R. G. Burns ◽  
Yuri Medvedev
Keyword(s):  
Large Class ◽  
Finite Groups ◽  
Additional Condition ◽  
Metabelian Group ◽  
Nilpotency Class ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Locally Graded Groups ◽  
Finite Exponent ◽  
Group Law

AbstractIf ω ≡ 1 is a group law implying virtual nilpotence in every finitely generated metabelian group satisfying it, then it implies virtual nilpotence for the finitely generated groups of a large class of groups including all residually or locally soluble-or-finite groups. In fact the groups of satisfying such a law are all nilpotent-by-finite exponent where the nilpotency class and exponent in question are both bounded above in terms of the length of ω alone. This yields a dichotomy for words. Finally, if the law ω ≡ 1 satisfies a certain additional condition—obtaining in particular for any monoidal or Engel law—then the conclusion extends to the much larger class consisting of all ‘locally graded’ groups.

scholarly journals Locally graded groups with all subgroups normal-by-finite

1996 ◽  
Vol 60 (2) ◽  
pp. 222-227 ◽  
Author(s):  
Howard Smith ◽  
James Wiegold
Keyword(s):  
Finite Group ◽  
Finite Index ◽  
Finitely Generated ◽  
Locally Finite ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Open Question ◽  
Locally Graded Groups

AbstractIn a paper published in this journal [1], J. T. Buckely, J. C. Lennox, B. H. Neumann and the authors considered the class of CF-groups, that G such that |H: CoreG (H)| is finite for all subgroups H. It is shown that locally finite CF-groups are abelian-by-finite and BCF, that is, there is an integer n such that |H: CoreG(H)| ≤ n for all subgroups H. The present paper studies these properties in the class of locally graded groups, the main result being that locally graded BCF-groups are abelian-by-finite. Whether locally graded CF-groups are BFC remains an open question. In this direction, the following problems is posed. Does there exist a finitely generated infinite periodic residually finite group in which all subgroups are finite or of finite index? Such groups are locally graded and CF but not BCF.

scholarly journals On Positive Law Problems in the Class of Locally Graded Groups

2004 ◽  
Vol 32 (5) ◽  
pp. 1841-1846 ◽  
Author(s):  
B. Bajorska ◽  
O. Macedońska
Keyword(s):  
Positive Law ◽  
Locally Graded Groups

Groups with restricted non-permutable subgroups

2020 ◽  
pp. 2150062
Author(s):  
Fausto De Mari
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Minimal Condition ◽  
Finitely Generated ◽  
Derived Length ◽  
Permutable Subgroups ◽  
Locally Graded Groups

A subgroup [Formula: see text] of a group [Formula: see text] is said to be permutable if [Formula: see text] for every subgroup [Formula: see text] of [Formula: see text] and the group [Formula: see text] is called metaquasihamiltonian if all subgroups of [Formula: see text] are either permutable or abelian. It is known that a locally graded metaquasihamiltonian group [Formula: see text] is soluble with derived length at most [Formula: see text] and contains a finite normal subgroup [Formula: see text] such that all subgroups of the factor [Formula: see text] are permutable. In this paper, we describe locally graded groups in which the set of all nonmetaquasihamiltonian subgroups satisfies the minimal condition and locally graded groups with the minimal condition on subgroups which are neither abelian nor permutable. Moreover, it is proved here that a finitely generated hyper-(abelian or finite) group whose finite homomorphic images are metaquasihamiltonian is itself metaquasihamiltonian.

scholarly journals Locally graded groups with a nilpotency condition on infinite subsets

2000 ◽  
Vol 69 (3) ◽  
pp. 415-420 ◽  
Author(s):  
Costantino Delizia ◽  
Akbar Rhemtulla ◽  
Howard Smith
Keyword(s):  
Positive Integer ◽  
Infinite Subset ◽  
Nilpotent Subgroup ◽  
Finitely Generated ◽  
Finite Image ◽  
Nontrivial Subgroup ◽  
Locally Graded Groups

AbstractA group G is locally graded if every finitely generated nontrivial subgroup of G has a nontrivial finite image. Let N (2, k)* denote the class of groups in which every infinite subset contains a pair of elements that generate a nilpotent subgroup of class at most k. We show that if G is a finitely generated locally graded N (2, k)*-group, then there is a positive integer c depending only on k such that G/Zc (G) is finite.

ON GROUPS WHICH CONTAIN NO HNN-EXTENSIONS

2007 ◽  
Vol 17 (07) ◽  
pp. 1377-1387 ◽  
Author(s):  
DEREK J. S. ROBINSON ◽  
ALESSIO RUSSO ◽  
GIOVANNI VINCENZI
Keyword(s):  
Large Class ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Hnn Extensions ◽  
Rank 2 ◽  
Locally Graded Groups

A group is called HNN-free if it has no subgroups that are nontrivial HNN-extensions. We prove that finitely generated HNN-free implies virtually polycyclic for a large class of groups. We also consider finitely generated groups with no free subsemigroups of rank 2 and show that in many situations such groups are virtually nilpotent. Finally, as an application of our results, we determine the structure of locally graded groups in which every subgroup is pronormal, thus generalizing a theorem of Kuzennyi and Subbotin.

Groups with Finiteness Conditions on the Lower Central Series of Subgroups

2013 ◽  
Vol 20 (04) ◽  
pp. 663-670 ◽  
Author(s):  
Silvana Rinauro
Keyword(s):  
Positive Integer ◽  
Lower Central Series ◽  
Central Series ◽  
Finiteness Conditions ◽  
Locally Graded Groups

Let k be a positive integer. Locally graded groups G for which one of the sets {γk(H)| H ≤ G} and {γk(H)| H ≤G, H infinite } is finite are classified.

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