Groups Whose Subgroup Growth is Less than Linear

1997 ◽  
Vol 07 (01) ◽  
pp. 77-91 ◽  
Author(s):  
Aner Shalev
Keyword(s):  
Finite Group ◽  
Finite Index ◽  
Residually Finite ◽  
Subgroup Growth ◽  
Residually Finite Group ◽  
Finite Quotients

Let G be a residually finite group and let an(G) denote the number of index n subgroups of G. It is shown that an(G)/n →0 if and only if G has a finite index central subgroup whose finite quotients are all cyclic. As an application we show that the degree of a group of polynomial subgroup growth cannot lie strictly between 0 and 1.

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.

A property of finitely generated residually finite groups

1976 ◽  
Vol 15 (3) ◽  
pp. 347-350 ◽  
Author(s):  
P.F. Pickel
Keyword(s):  
Finite Group ◽  
Homomorphic Image ◽  
Finite Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Isomorphism Classes ◽  
Residually Finite Groups ◽  
Finite Quotients ◽  
Relatively Free Group

Let F(G) denote the set of isomorphism classes of finite quotients of the group G. We say that groups G and H have isomorphic finite quotients (IFQ) if F(G) = F(H). In this note, we show that a finitely generated residually finite group G cannot have the same finite quotients as a proper homomorphic image (G is IFQ hopfian). We then obtain some results on groups with the same finite quotients as a relatively free group.

scholarly journals APPROXIMATING THE FIRST L2-BETTI NUMBER OF RESIDUALLY FINITE GROUPS

2011 ◽  
Vol 03 (02) ◽  
pp. 153-160 ◽  
Author(s):  
W. LÜCK ◽  
D. OSIN
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Betti Number ◽  
Betti Numbers ◽  
Torsion Group ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Residually Finite Groups

We show that the first L2-betti number of a finitely generated residually finite group can be estimated from below by using ordinary first betti numbers of finite index normal subgroups. As an application, we construct a finitely generated infinite residually finite torsion group with positive first L2-betti number.

Residually finite groups of finite rank

1989 ◽  
Vol 106 (3) ◽  
pp. 385-388 ◽  
Author(s):  
Alexander Lubotzky ◽  
Avinoam Mann
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Finite Rank ◽  
Prime Order ◽  
Residual Finiteness ◽  
Infinite Groups ◽  
Finiteness Conditions ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Residually Finite Groups

The recent constructions, by Rips and Olshanskii, of infinite groups with all proper subgroups of prime order, and similar ‘monsters’, show that even under the imposition of apparently very strong finiteness conditions, the structure of infinite groups can be rather weird. Thus it seems reasonable to impose the type of condition that enables us to apply the theory of finite groups. Two such conditions are local finiteness and residual finiteness, and here we are interested in the latter. Specifically, we consider residually finite groups of finite rank, where a group is said to have rank r, if all finitely generated subgroups of it can be generated by r elements. Recall that a group is said to be virtually of some property, if it has a subgroup of finite index with this property. We prove the following result:Theorem 1. A residually finite group of finite rank is virtually locally soluble.

scholarly journals ON GROUPS ADMITTING A WORD WHOSE VALUES ARE ENGEL

2013 ◽  
Vol 23 (01) ◽  
pp. 81-89 ◽  
Author(s):  
RAIMUNDO BASTOS ◽  
PAVEL SHUMYATSKY ◽  
ANTONIO TORTORA ◽  
MARIA TOTA
Keyword(s):  
Finite Group ◽  
Multilinear Commutator ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Verbal Subgroup ◽  
Locally Nilpotent ◽  
Locally Graded Group ◽  
Commutator Word ◽  
Positive Integers

Let m, n be positive integers, v a multilinear commutator word and w = vm. We prove that if G is a residually finite group in which all w-values are n-Engel, then the verbal subgroup w(G) is locally nilpotent. We also examine the question whether this is true in the case where G is locally graded rather than residually finite. We answer the question affirmatively in the case where m = 1. Moreover, we show that if u is a non-commutator word and G is a locally graded group in which all u-values are n-Engel, then the verbal subgroup u(G) is locally nilpotent.

NILPOTENT LINEAR SEMIGROUPS

2006 ◽  
Vol 16 (01) ◽  
pp. 141-160 ◽  
Author(s):  
ERIC JESPERS ◽  
DAVID RILEY
Keyword(s):  
Finite Group ◽  
Lie Algebra ◽  
Associative Algebra ◽  
Burnside Problem ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Adjoint Semigroup ◽  
Restricted Burnside Problem ◽  
Global And Local

We characterize the structure of linear semigroups satisfying certain global and local nilpotence conditions and deduce various Engel-type results. For example, using a form of Zel'manov's solution of the restricted Burnside problem we are able to show that a finitely generated residually finite group is nilpotent if and only if it satisfies a certain 4-generator property of semigroups we call WMN. Methods of linear semigroups then allow us to prove that a linear semigroup is Mal'cev nilpotent precisely when it satisfies WMN. As an application, we show that a finitely generated associative algebra is nilpotent when viewed as a Lie algebra if and only if its adjoint semigroup is WMN.

ELEMENTS OF PRIME POWER ORDER IN RESIDUALLY FINITE GROUPS

2005 ◽  
Vol 15 (03) ◽  
pp. 571-576 ◽  
Author(s):  
PAVEL SHUMYATSKY
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Prime Power Order ◽  
Closed Set ◽  
Locally Finite ◽  
Residually Finite ◽  
Residually Finite Group ◽  
P Elements ◽  
Residually Finite Groups

Let G be a residually finite group satisfying some identity w ≡ 1. Suppose G is generated by a normal commutator-closed set X of p-elements. We prove that G is locally finite.

scholarly journals Some properties of endomorphisms in residually finite groups

1977 ◽  
Vol 24 (1) ◽  
pp. 117-120 ◽  
Author(s):  
Ronald Hirshon
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Free Product ◽  
Finite Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Residually Finite Groups ◽  
A Finite Group ◽  
Torsion Free

AbstractIf ε is an endomorphism of a finitely generated residually finite group onto a subgroup Fε of finite index in F, then there exists a positive integer k such that ε is an isomorphism of Fεk. If K is the kernel of ε, then K is a finite group so that if F is a non trivial free product or if F is torsion free, then ε is an isomorphism on F. If ε is an endomorphism of a finitely generated resedually finite group onto a subgroup Fε (not necessatily of ginite index in F) and if the kernel of ε obeys the minimal condition for subgroups then there exists a positive integer k such that ε is an isomorphism on Fεk.

scholarly journals Computability of finite quotients of finitely generated groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Emmanuel Rauzy
Keyword(s):  
Word Problem ◽  
Study Groups ◽  
The Other ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Finitely Generated Groups ◽  
Solvable Word Problem ◽  
Finite Quotients ◽  
Solvable Word ◽  
Recursive Set

Abstract We systematically study groups whose marked finite quotients form a recursive set. We give several definitions, and prove basic properties of this class of groups, and in particular emphasize the link between the growth of the depth function and solvability of the word problem. We give examples of infinitely presented groups whose finite quotients can be effectively enumerated. Finally, our main result is that a residually finite group can fail to be recursively presented and still have computable finite quotients, and that, on the other hand, it can have solvable word problem but not have computable finite quotients.

scholarly journals Obstruction to a Higman embedding theorem for residually finite groups with solvable word problem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Emmanuel Rauzy
Keyword(s):  
Finite Group ◽  
Word Problem ◽  
Identity Element ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Residually Finite Groups ◽  
Solvable Word Problem ◽  
A Finite Group ◽  
Finitely Presented ◽  
Solvable Word

AbstractWe prove that, for a finitely generated residually finite group, having solvable word problem is not a sufficient condition to be a subgroup of a finitely presented residually finite group. The obstruction is given by a residually finite group with solvable word problem for which there is no effective method that allows, given some non-identity element, to find a morphism onto a finite group in which this element has a non-trivial image. We also prove that the depth function of this group grows faster than any recursive function.

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