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.

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.

scholarly journals Some properties of the growth and of the algebraic entropy of group endomorphisms

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Anna Giordano Bruno ◽  
Pablo Spiga
Keyword(s):  
Finite Groups ◽  
Addition Theorem ◽  
Finitely Generated ◽  
Growth Type ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Algebraic Entropy ◽  
Finitely Generated Groups ◽  
Identity Automorphism ◽  
Inner Automorphisms

AbstractWe study the growth of group endomorphisms, a generalization of the classical notion of growth of finitely generated groups, which is strictly related to algebraic entropy. We prove that the inner automorphisms of a group have the same growth type and the same algebraic entropy as the identity automorphism. Moreover, we show that endomorphisms of locally finite groups cannot have intermediate growth. We also find an example showing that the Addition Theorem for algebraic entropy does not hold for endomorphisms of arbitrary groups.

scholarly journals Infinite presentability of groups and condensation

2014 ◽  
Vol 13 (4) ◽  
pp. 811-848 ◽  
Author(s):  
Robert Bieri ◽  
Yves Cornulier ◽  
Luc Guyot ◽  
Ralph Strebel
Keyword(s):  
Metabelian Group ◽  
Finitely Generated ◽  
Finitely Generated Groups

AbstractWe describe various classes of infinitely presented groups that are condensation points in the space of marked groups. A well-known class of such groups consists of finitely generated groups admitting an infinite minimal presentation. We introduce here a larger class of condensation groups, called infinitely independently presentable groups, and establish criteria which allow one to infer that a group is infinitely independently presentable. In addition, we construct examples of finitely generated groups with no minimal presentation, among them infinitely presented groups with Cantor–Bendixson rank 1, and we prove that every infinitely presented metabelian group is a condensation group.

scholarly journals On quasinormal subgroups of certain finitely generated groups

1983 ◽  
Vol 26 (1) ◽  
pp. 25-28 ◽  
Author(s):  
John C. Lennox
Keyword(s):  
Nilpotent Group ◽  
Normal Closure ◽  
Infinite Index ◽  
Finitely Generated ◽  
The Core ◽  
Finitely Generated Groups ◽  
Quasinormal Subgroup ◽  
Finite Exponent

A subgroup Q of a group G is called quasinormal in G if Q permutes with every subgroup of G. Of course a quasinormal subgroup Q of a group G may be very far from normal. In fact, examples of Iwasawa show (for a convenient reference see [8]) that we may have Q core-free and the normal closure QG of Q in G equal to G so that Q is not even subnormal in G. We note also that the core of Q in G, QG, is of infinite index in QG in this example. If G is finitely generated then any quasinormal subgroup Q is subnormal in G [8] and although Q is not necessarily normal in G we have that |QG:Q| is finite and |QG:Q| is a nilpotent group of finite exponent [5].

Faithful Representations of Finitely Generated Metabelian Groups

1975 ◽  
Vol 27 (6) ◽  
pp. 1355-1360 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Finite Index ◽  
Characteristic Zero ◽  
Metabelian Group ◽  
Finite Extension ◽  
Faithful Representation ◽  
Finitely Generated ◽  
Finite Degree ◽  
Metabelian Groups ◽  
Finite Exponent ◽  
Torsion Free

In [3] Remeslennikov proves that a finitely generated metabelian group G has a faithful representation of finite degree over some field F of characteristic zero (respectively, p > 0) if its derived group G’ is torsion-free (respectively, of exponent p). By the Lie-Kolchin-Mal'cev theorem any metabelian subgroup of GL(n, F) has a subgroup of finite index whose derived group is torsion-free if char F = 0 and is a p-group of finite exponent if char F = p > 0. Moreover every finite extension of a group with a faithful representation (of finite degree) has a faithful representation over the same field. Thus Remeslennikov's results have a gap which we propose here to fill.

scholarly journals On the residual and profinite closures of commensurated subgroups

2019 ◽  
Vol 169 (2) ◽  
pp. 411-432
Author(s):  
PIERRE–EMMANUEL CAPRACE ◽  
PETER H. KROPHOLLER ◽  
COLIN D. REID ◽  
PHILLIP WESOLEK
Keyword(s):  
Finite Groups ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Infinite Groups ◽  
Residually Finite ◽  
Finitely Generated Groups ◽  
Residually Finite Groups ◽  
Groups Acting On Trees ◽  
Polycyclic Groups

AbstractThe residual closure of a subgroup H of a group G is the intersection of all virtually normal subgroups of G containing H. We show that if G is generated by finitely many cosets of H and if H is commensurated, then the residual closure of H in G is virtually normal. This implies that separable commensurated subgroups of finitely generated groups are virtually normal. A stream of applications to separable subgroups, polycyclic groups, residually finite groups, groups acting on trees, lattices in products of trees and just-infinite groups then flows from this main result.

Direct Cancellation of Finite Groups

2005 ◽  
Vol 12 (04) ◽  
pp. 563-566
Author(s):  
Andrew Fransman ◽  
Peter Witbooi
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
A Finite Group

We prove that if F is a finite group, and G and H are finitely generated groups with finite commutator subgroups for which F × G ≃ F × H, then G ≃ H.

scholarly journals The subnormal coalescence of some classes of groups of finite rank

1973 ◽  
Vol 16 (3) ◽  
pp. 324-327 ◽  
Author(s):  
Mark Drukker ◽  
Derek J. S. Robinson ◽  
Ian Stewart
Keyword(s):  
Finite Groups ◽  
Finite Rank ◽  
Nilpotent Groups ◽  
Subnormal Subgroup ◽  
Minimal Condition ◽  
Finitely Generated ◽  
Maximal Condition ◽  
Finitely Generated Groups ◽  
Subnormal Subgroups

A class of groups forms a (subnormal) coalition class, or is (subnormally) coalescent, if wheneverHandKare subnormal -subgroups of a groupGthen their join <H, K> is also a subnormal -subgroup ofG. Among the known coalition classes are those of finite groups and polycylic groups (Wielandt [15]); groups with maximal condition for subgroups (Baer [1]); finitely generated nilpotent groups (Baer [2]); groups with maximal or minimal condition on subnormal subgroups (Robinson [8], Roseblade [11, 12]); minimax groups (Roseblade, unpublished); and any subjunctive class of finitely generated groups (Roseblade and Stonehewer [13]).

scholarly journals Embedding group amalgams

1981 ◽  
Vol 24 (3) ◽  
pp. 373-379
Author(s):  
Patrick Fitzpatrick ◽  
James Wiegold
Keyword(s):  
Normal Subgroup ◽  
Finite Groups ◽  
Abelian Varieties ◽  
Finitely Generated ◽  
Large Classes ◽  
Nilpotent Variety ◽  
Finitely Generated Groups

A class of groups is said to have the small embedding property if every amalgam of two groups from amalgamating a normal subgroup embeds into a group not generating the variety of all groups. Very few large classes have the small embedding property. For example, the minimal non-abelian varieties containing non-abelian finite groups do not; neither does any class of groups containing ; but the set of finitely generated groups in a nilpotent variety not containing does have the small embedding property.

scholarly journals Condensed groups in product varieties

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Denis Osin
Keyword(s):  
Isomorphism Class ◽  
Product Variety ◽  
Equivalence Relations ◽  
Finitely Generated ◽  
Locally Finite ◽  
Elementary Equivalence ◽  
Finitely Generated Groups ◽  
Locally Finite Variety ◽  
Isolated Points ◽  
Finite Exponent

Abstract A finitely generated group 𝐺 is said to be condensed if its isomorphism class in the space of finitely generated marked groups has no isolated points. We prove that every product variety U ⁢ V \mathcal{UV} , where 𝒰 (respectively, 𝒱) is a non-abelian (respectively, a non-locally finite) variety, contains a condensed group. In particular, there exist condensed groups of finite exponent. As an application, we obtain some results on the structure of the isomorphism and elementary equivalence relations on the set of finitely generated groups in U ⁢ V \mathcal{UV} .

Sign in / Sign up

Export Citation Format

Share Document