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 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.

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 A question of Paul Erdös and nilpotent-by-finite groups

2001 ◽  
Vol 64 (2) ◽  
pp. 245-254
Author(s):  
Bijan Taeri
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
Soluble Group ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Finite Quotient

Let n be a positive integer or infinity (denoted ∞), k a positive integer. We denote by Ωk(n) the class of groups G such that, for every subset X of G of cardinality n + 1, there exist distinct elements x, y ∈ X and integers t0, t1…, tk such that , where xi, ∈ {x, y}, i = 0, 1,…,k, x0 ≠ x1. If the integers t0, t1,…,tk are the same for any subset X of G, we say that G is in the class Ω̅k(n). The class k (n) is defined exactly as Ωk(n) with the additional conditions . Let t2, t3,…,tk be fixed integers. We denote by the class of all groups G such that for any infinite subsets X and Y there exist x ∈ X, y ∈ Y such that , where xi ∈ {x, y}, x0 ≠ x1, i = 2, 3, …, k. Here we prove that (1) If G ∈ k(2) is a finitely generated soluble group, then G is nilpotent.(2) If G ∈ Ωk(∞) is a finitely generated soluble group, then G is nilpotentby-finite.(3) If G ∈ Ω̅k(n), n a positive integer, is a finitely generated residually finite group, then G is nilpotent-by-finite.(4) If G is an infinite -group in which every nontrivial finitely generated subgroup has a nontrivial finite quotient, then G is nilpotent-by-finite.

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.

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 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.

scholarly journals ON LOCAL FINITENESS OF PERIODIC RESIDUALLY FINITE GROUPS

2002 ◽  
Vol 45 (3) ◽  
pp. 717-721 ◽  
Author(s):  
Mahmut Kuzucuoğlu ◽  
Pavel Shumyatsky
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Nilpotent Subgroup ◽  
Subject Classification ◽  
Locally Finite ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Local Finiteness ◽  
Residually Finite Groups

AbstractLet $G$ be a periodic residually finite group containing a nilpotent subgroup $A$ such that $C_G(A)$ is finite. We show that if $\langle A,A^g\rangle$ is finite for any $g\in G$, then $G$ is locally finite.AMS 2000 Mathematics subject classification: Primary 20F50

scholarly journals ON VERBAL SUBGROUPS IN RESIDUALLY FINITE GROUPS

2011 ◽  
Vol 84 (1) ◽  
pp. 159-170 ◽  
Author(s):  
JHONE CALDEIRA ◽  
PAVEL SHUMYATSKY
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Locally Finite ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Verbal Subgroup ◽  
Residually Finite Groups ◽  
Positive Integers ◽  
Verbal Subgroups

AbstractThe following theorem is proved. Let m, k and n be positive integers. There exists a number η=η(m,k,n) depending only on m, k and n such that if G is any residually finite group satisfying the condition that the product of any η commutators of the form [xm,y1,…,yk ] is of order dividing n, then the verbal subgroup of G corresponding to the word w=[xm,y1,…,yk ] is locally finite.

scholarly journals An uncountable family of finitely generated residually finite groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hip Kuen Chong ◽  
Daniel T. Wise
Keyword(s):  
Free Group ◽  
Finite Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Uncountable Family ◽  
Residually Finite Groups ◽  
Rank 2

Abstract We study a family of finitely generated residually finite groups. These groups are doubles F 2 * H F 2 F_{2}*_{H}F_{2} of a rank-2 free group F 2 F_{2} along an infinitely generated subgroup 𝐻. Varying 𝐻 yields uncountably many groups up to isomorphism.

scholarly journals Groups whose Chermak–Delgado lattice is a quasi-antichain

2019 ◽  
Vol 22 (3) ◽  
pp. 529-544
Author(s):  
Lijian An
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
A Finite Group

Abstract A quasi-antichain is a lattice consisting of a maximum, a minimum, and the atoms of the lattice. The width of a quasi-antichain is the number of atoms. For a positive integer {w\geq 3} , a quasi-antichain of width w is denoted by {\mathcal{M}_{w}} . In [B. Brewster, P. Hauck and E. Wilcox, Quasi-antichain Chermak–Delgado lattice of finite groups, Arch. Math. 103 2014, 4, 301–311], it is proved that {\mathcal{M}_{w}} can be the Chermak–Delgado lattice of a finite group if and only if {w=1+p^{a}} for some positive integer a and some prime p. Let t be the number of abelian atoms in {\mathcal{CD}(G)} . In this paper, we completely answer the following question: which values of t are possible in quasi-antichain Chermak–Delgado lattices?

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