Tarski's problem for varieties of groups with a commutator identity

1986 ◽  
Vol 51 (1) ◽  
pp. 75-78
Author(s):  
John Lawrence
Keyword(s):  
Finite Rank ◽  
Free Groups ◽  
Commutator Identity ◽  
Variety Of Groups

AbstractIt is proved that for a variety of groups in which the relatively free groups are solvable, the relatively free groups of distinct finite rank are not elementarily equivalent.

scholarly journals Generating groups of certain soluble varieties

1974 ◽  
Vol 17 (2) ◽  
pp. 222-233 ◽  
Author(s):  
Narain Gupta ◽  
Frank Levin
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
Free Groups ◽  
Infinite Rank ◽  
Variety Of Groups ◽  
Countably Infinite

Any variety of groups is generated by its free group of countably infinite rank. A problem that appears in various forms in Hanna Neumann's book [7] (see, for intance, sections 2.4, 2.5, 3.5, 3.6) is that of determining if a given variety B can be generated by Fk(B), one of its free groups of finite rank; and if so, if Fn(B) is residually a k-generator group for all n ≧ k. (Here, as in the sequel, all unexplained notation follows [7].)

scholarly journals On the generic type of the free group

2011 ◽  
Vol 76 (1) ◽  
pp. 227-234 ◽  
Author(s):  
Rizos Sklinos
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
Free Groups ◽  
Primitive Elements ◽  
Generic Type

AbstractWe answer a question raised in [9], that is whether the infinite weight of the generic type of the free group is witnessed in Fω. We also prove that the set of primitive elements in finite rank free groups is not uniformly definable. As a corollary, we observe that the generic type over the empty set is not isolated. Finally, we show that uncountable free groups are not ℵ1-homogeneous.

On Torsion-Free Groups of Finite Rank

1984 ◽  
Vol 36 (6) ◽  
pp. 1067-1080 ◽  
Author(s):  
David Meier ◽  
Akbar Rhemtulla
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Finite Rank ◽  
Maximal Element ◽  
Free Groups ◽  
Solvable Groups ◽  
Finitely Generated ◽  
Isolated Subgroup ◽  
Image Position ◽  
Torsion Free

This paper deals with two conditions which, when stated, appear similar, but when applied to finitely generated solvable groups have very different effect. We first establish the notation before stating these conditions and their implications. If H is a subgroup of a group G, let denote the setWe say G has the isolator property if is a subgroup for all H ≦ G. Groups possessing the isolator property were discussed in [2]. If we define the relation ∼ on the set of subgroups of a given group G by the rule H ∼ K if and only if , then ∼ is an equivalence relation and every equivalence class has a maximal element which may not be unique. If , we call H an isolated subgroup of G.

Approximability of outer automorphism groups of free groups of finite rank

1981 ◽  
Vol 20 (3) ◽  
pp. 194-200
Author(s):  
A. G. Myasnikov
Keyword(s):  
Finite Rank ◽  
Automorphism Groups ◽  
Free Groups ◽  
Outer Automorphism ◽  
Outer Automorphism Groups

AUTOMORPHISMS OF RELATIVELY FREE GROUPS IN THE VARIETY N2A ∧ AN2 ∧ Nc

2001 ◽  
Vol 63 (3) ◽  
pp. 607-622 ◽  
Author(s):  
ATHANASSIOS I. PAPISTAS
Keyword(s):  
Automorphism Group ◽  
Free Group ◽  
Finite Index ◽  
Finite Rank ◽  
Free Groups ◽  
Finite Set ◽  
Tame Automorphism ◽  
Positive Integers ◽  
Relatively Free Group

For positive integers n and c, with n [ges ] 2, let Gn, c be a relatively free group of finite rank n in the variety N2A ∧ AN2 ∧ Nc. It is shown that the subgroup of the automorphism group Aut(Gn, c) of Gn, c generated by the tame automorphisms and an explicitly described finite set of IA-automorphisms of Gn, c has finite index in Aut(Gn, c). Furthermore, it is proved that there are no non-trivial elements of Gn, c fixed by every tame automorphism of Gn, c.

scholarly journals The Grushko decomposition of a finite graph of finite rank free groups: an algorithm

2005 ◽  
Vol 9 (4) ◽  
pp. 1835-1880 ◽  
Author(s):  
Guo-An Diao ◽  
Mark Feighn
Keyword(s):  
Finite Rank ◽  
Free Groups ◽  
Finite Graph

scholarly journals The Herzog–Schönheim conjecture for finitely generated groups

2019 ◽  
Vol 29 (06) ◽  
pp. 1083-1112 ◽  
Author(s):  
Fabienne Chouraqui
Keyword(s):  
Metric Space ◽  
Finite Rank ◽  
Free Groups ◽  
Sufficient Conditions ◽  
Combinatorial Approach ◽  
Finitely Generated ◽  
Covering Spaces ◽  
Finitely Generated Groups

Let [Formula: see text] be a group and [Formula: see text] be subgroups of [Formula: see text] of indices [Formula: see text], respectively. In 1974, Herzog and Schönheim conjectured that if [Formula: see text], [Formula: see text], is a coset partition of [Formula: see text], then [Formula: see text] cannot be distinct. We consider the Herzog–Schönheim conjecture for free groups of finite rank and develop a new combinatorial approach, using covering spaces. We define [Formula: see text] the space of coset partitions of [Formula: see text] and show [Formula: see text] is a metric space with interesting properties. We give some sufficient conditions on the coset partition that ensure the conjecture is satisfied and moreover has a neighborhood [Formula: see text] in [Formula: see text] such that all the partitions in [Formula: see text] satisfy also the conjecture.

scholarly journals Faithful linear representations of certain free nilpotent groups

1995 ◽  
Vol 37 (1) ◽  
pp. 33-36 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Free Groups ◽  
Linear Representation ◽  
Nilpotent Groups ◽  
Finite Degree ◽  
Final Comment ◽  
Linear Representations ◽  
Variety Of Groups ◽  
Countable Rank ◽  
Faithful Linear Representation ◽  
Finitary Linear

Brian Hartley asked me whether a free (nilpotent of class 2 and exponent p2)-group of countable rank has a faithful linear representation of finite degree, p here being a prime of course. The answer is yes. The point is that this then yields via work of F. Leinen and M. J. Tomkinson, see [3,3.6] an image of a linear p-group, which is not even finitary linear. The question of which relatively free groups have faithful linear representations dates back at least to work of W. Magnus in the 1930's, see [4, pp. 33, 34 and the final comment on p. 40] for a discussion of this. Our construction, which works more generally, is a further contribution. We write ℜc for the variety of nilpotent groups of class at most c and (ℭ9 for the variety of groups of exponent dividing q.

COMMUTATOR SUBGROUPS OF FREE NILPOTENT GROUPS

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1021-1031
Author(s):  
N. GUPTA ◽  
I. B. S. PASSI
Keyword(s):  
Nilpotent Group ◽  
Free Group ◽  
Finite Rank ◽  
Free Groups ◽  
Nilpotent Groups ◽  
Finite Chain ◽  
Free Nilpotent Group ◽  
Infinite Rank ◽  
Derived Subgroup ◽  
Finite Set

For fixed m, n ≥ 2, we examine the structure of the nth lower central subgroup γn(F) of the free group F of rank m with respect to a certain finite chain F = F(0) > F(1) > ⋯ > F(l-1) > F(l) = {1} of free groups in which F(k) is of finite rank m(k) and is contained in the kth derived subgroup δk(F) of F. The derived subgroups δk(F/γn(F)) of the free nilpotent group F/γn(F) are isomorphic to the quotients F(k)/(F(k) ∩ γn(F)) and admit presentations of the form 〈xk,1,…,xk,m(k): γ(n)(F(k))〉, where γ(n)(F(k)), contained in γn(F), is a certain partial lower central subgroup of F(k). We give a complete description of γn(F) as a staggered product Π1 ≤ k ≤ l-1(γ〈n〉(F(k))*γ[n](F(k)))F(k+1), where γ〈n〉(F(k)) is a free factor of the derived subgroup [F(k),F(k)] of F(k) having countable infinite rank and generated by a certain set of reduced commutators of weight at least n, and γ[n](F(k)) is the subgroup generated by a certain finite set of products of non-reduced ordered commutators of weight at least n. There are some far-reaching consequences.

scholarly journals Inverse limits of finite rank free groups

2012 ◽  
Vol 15 (6) ◽  
Author(s):  
Gregory R. Conner ◽  
Curtis Kent
Keyword(s):  
Homomorphic Image ◽  
Free Group ◽  
Finite Rank ◽  
Inverse Limit ◽  
Free Groups ◽  
Inverse System ◽  
Inverse Limits ◽  
Universal Group ◽  
Hawaiian Earring ◽  
Constant System

Abstract.We will show that the inverse limit of finite rank free groups with surjective connecting homomorphism is isomorphic either to a finite rank free group or to a fixed universal group. In other words, any inverse system of finite rank free groups which is not equivalent to an eventually constant system has the universal group as its limit. This universal inverse limit is naturally isomorphic to the first shape group of the Hawaiian earring. We also give an example of a homomorphic image of the Hawaiian earring group which lies in the inverse limit of free groups but is neither a free group nor isomorphic to the Hawaiian earring group.

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