scholarly journals Large normal subgroup growth and large characteristic subgroup growth

2020 ◽  
Vol 23 (1) ◽  
pp. 1-15
Author(s):  
Yiftach Barnea ◽  
Jan-Christoph Schlage-Puchta
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Open Subgroup ◽  
Positive Answer ◽  
Characteristic Subgroup ◽  
Finitely Generated ◽  
Growth Type ◽  
Subgroup Growth ◽  
Finitely Generated Group ◽  
Standing Problem

AbstractThe fastest normal subgroup growth type of a finitely generated group is {n^{\log n}}. Very little is known about groups with this type of growth. In particular, the following is a long standing problem: Let Γ be a group and Δ a subgroup of finite index. Suppose Δ has normal subgroup growth of type {n^{\log n}}. Does Γ have normal subgroup growth of type {n^{\log n}}? We give a positive answer in some cases, generalizing a result of Müller and the second author and a result of Gerdau. For instance, suppose G is a profinite group and H an open subgroup of G. We show that if H is a generalized Golod–Shafarevich group, then G has normal subgroup growth of type {n^{\log n}}. We also use our methods to show that one can find a group with characteristic subgroup growth of type {n^{\log n}}.

A note on a centrality property of finitely generated soluble groups

1974 ◽  
Vol 75 (1) ◽  
pp. 23-24 ◽  
Author(s):  
J. C. Lennox
Keyword(s):  
Normal Subgroup ◽  
Positive Integer ◽  
Finite Index ◽  
Finitely Generated ◽  
Abelian Normal Subgroup ◽  
Soluble Groups ◽  
Finitely Generated Group ◽  
Torsion Free ◽  
Simpler Method

We recall from (3) that a group G is (centrally) eremitic if there exists a positive integer e such that, whenever an element of G has some power in a centralizer, it has its eth power. The eccentricity of an eremitic group G is the least such positive integer e.In ((4), Theorem A) we proved that if A is a torsion free Abelian normal subgroup of a finitely generated group G with G/A nilpotent, then G has a subgroup of finite index with eccentricity 1. In this note we use a much simpler method to prove a stronger result.

Second Fox Subgroups of Arbitrary Groups

1995 ◽  
Vol 38 (2) ◽  
pp. 177-181 ◽  
Author(s):  
Mario Curzio ◽  
C. Kanta Gupta
Keyword(s):  
Normal Subgroup ◽  
Finitely Generated ◽  
Finitely Generated Group

AbstractWe give a complete description of the second Fox subgroup G ∩ (1 + Δ2(G)Δ(H)) relative to a given normal subgroup H of an arbitrary finitely generated group G.

A Conjecture of Bachmuth and Mochizuki on Automorphisms of Soluble Groups

1976 ◽  
Vol 28 (6) ◽  
pp. 1302-1310 ◽  
Author(s):  
Brian Hartley
Keyword(s):  
Finite Index ◽  
Soluble Group ◽  
Abelian Groups ◽  
Free Subgroup ◽  
Linear Groups ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Group Of Automorphisms ◽  
Finitely Generated Group ◽  
Soluble Subgroup

In [1], Bachmuth and Mochizuki conjecture, by analogy with a celebrated result of Tits on linear groups [8], that a finitely generated group of automorphisms of a finitely generated soluble group either contains a soluble subgroup of finite index (which may of course be taken to be normal) or contains a non-abelian free subgroup. They point out that their conjecture holds for nilpotent-by-abelian groups and in some other cases.

scholarly journals On a theorem of Avez

2019 ◽  
Vol 22 (3) ◽  
pp. 383-395
Author(s):  
Murray Elder ◽  
Cameron Rogers
Keyword(s):  
Normal Subgroup ◽  
Probability Measure ◽  
Finitely Generated ◽  
Amenable Groups ◽  
Finitely Generated Group ◽  
Amenable Radical

Abstract For each symmetric, aperiodic probability measure μ on a finitely generated group G, we define a subset {A_{\mu}} consisting of group elements g for which the limit of the ratio {{\mu^{\ast n}(g)}/{\mu^{\ast n}(e)}} tends to 1. We prove that {A_{\mu}} is a subgroup, is amenable, contains every finite normal subgroup, and {G=A_{\mu}} if and only if G is amenable. For non-amenable groups we show that {A_{\mu}} is not always a normal subgroup and can depend on the measure. We formulate some conjectures relating {A_{\mu}} to the amenable radical.

scholarly journals ON GROUPS WHOSE GEODESIC GROWTH IS POLYNOMIAL

2012 ◽  
Vol 22 (05) ◽  
pp. 1250048 ◽  
Author(s):  
MARTIN R. BRIDSON ◽  
JOSÉ BURILLO ◽  
MURRAY ELDER ◽  
ZORAN ŠUNIĆ
Keyword(s):  
Nilpotent Group ◽  
Finite Index ◽  
The Other ◽  
Normal Closure ◽  
Finitely Generated ◽  
Generating Set ◽  
Finitely Generated Group ◽  
Cyclic Groups ◽  
Growth Functions ◽  
Generating Sets

This paper records some observations concerning geodesic growth functions. If a nilpotent group is not virtually cyclic then it has exponential geodesic growth with respect to all finite generating sets. On the other hand, if a finitely generated group G has an element whose normal closure is abelian and of finite index, then G has a finite generating set with respect to which the geodesic growth is polynomial (this includes all virtually cyclic groups).

A REMARK ABOUT COMBINGS OF GROUPS

1993 ◽  
Vol 03 (04) ◽  
pp. 575-581 ◽  
Author(s):  
MARTIN R. BRIDSON ◽  
ROBERT H. GILMAN
Keyword(s):  
Finite Index ◽  
Free Subgroup ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Analogous Question

It is currently unknown whether or not there exist (synchronously) combable groups which are not automatic. This note settles an analogous question in a simpler context. The notion of a broomlike combing is introduced. It is shown that if a group admits a broomlike combing then it admits a broomlike combing in which the language of combing words is regular. A finitely generated group admits a broomlike combing if and only if it contains a free subgroup of finite index.

A Note on the Hopficity of Finitely Generated Free-by-Nilpotent Groups

1977 ◽  
Vol 20 (1) ◽  
pp. 33-34
Author(s):  
James Boler
Keyword(s):  
Normal Subgroup ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Finitely Generated Group

Our purpose is to deduce from a theorem of P. Hall the following observation.Let G be a finitely generated group and F a free normal subgroup of G with G/F nilpotent. Then G is hopfian.Here a group G is hopfian if every epiendomorphism G→G is an automorphism.

scholarly journals From local to global conjugacy of subgroups of relatively hyperbolic groups

2017 ◽  
Vol 27 (03) ◽  
pp. 299-314
Author(s):  
Oleg Bogopolski ◽  
Kai-Uwe Bux
Keyword(s):  
Finite Index ◽  
Minimal Length ◽  
Hyperbolic Groups ◽  
Index Subgroup ◽  
Finitely Generated ◽  
Limit Group ◽  
Relatively Hyperbolic Groups ◽  
Finitely Generated Group ◽  
Finite Index Subgroup ◽  
Relatively Hyperbolic

Suppose that a finitely generated group [Formula: see text] is hyperbolic relative to a collection of subgroups [Formula: see text]. Let [Formula: see text] be subgroups of [Formula: see text] such that [Formula: see text] is relatively quasiconvex with respect to [Formula: see text] and [Formula: see text] is not parabolic. Suppose that [Formula: see text] is elementwise conjugate into [Formula: see text]. Then there exists a finite index subgroup of [Formula: see text] which is conjugate into [Formula: see text]. The minimal length of the conjugator can be estimated. In the case, where [Formula: see text] is a limit group, it is sufficient to assume only that [Formula: see text] is a finitely generated and [Formula: see text] is an arbitrary subgroup of [Formula: see text].

ON SOME TYPES OF NOETHERIAN MODULES

2004 ◽  
Vol 03 (02) ◽  
pp. 169-179 ◽  
Author(s):  
LEONID A. KURDACHENKO ◽  
IGOR YA. SUBBOTIN
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Dedekind Domain ◽  
Finitely Generated ◽  
Abelian Normal Subgroup ◽  
Soluble Groups ◽  
Noetherian Module ◽  
Factor Module ◽  
Proper Factor ◽  
Noetherian Modules

We describe FC-hypercentral locally soluble groups G such that a groups algebra DG over a Dedekind domain D has a noetherian module A with faithful action of G such that A is subdirectly reducible and every proper factor module of A is finitely generated over D. It is proved, in particular, that G has an abelian normal subgroup of finite index.

scholarly journals Tameness and geodesic cores of subgroups

2000 ◽  
Vol 69 (2) ◽  
pp. 153-161 ◽  
Author(s):  
Rita Gitik
Keyword(s):  
Normal Subgroup ◽  
Factor Group ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Negatively Curved ◽  
Quasiconvex Subgroups ◽  
Trivial Subgroup

AbstractLet N be a finitely generated normal subgroup of a finitely generated group G. We show that if the trivial subgroup is tame in the factor group G/N, then N is that in G. We also give a short new proof of the fact that quasiconvex subgroups of negatively curved groups are tame. The proof utilizes the concept of the geodesic core of the subgroup and is related to the Dehn algorithm.

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