scholarly journals Some infinite Fibonacci groups

1975 ◽  
Vol 19 (3) ◽  
pp. 311-314 ◽  
Author(s):  
D. L. Johnson
Keyword(s):  
Normal Subgroup ◽  
Natural Number ◽  
Free Group ◽  
Factor Group ◽  
Image Position ◽  
Special Case

The Fibonacci groups are a special case of the following class of groups first studied by G. A. Miller (7). Given a natural number n, let θ be the automorphism of the free group F = 〈x1, …, xn |〉 of rank n which permutes the subscripts of the generators in accordance with the cycle (1, 2, …, n). Given a word w in F, let R be the smallest normal subgroup of F which contains w and is closed under θ. Then define Gn(w) = F/R and write An(w) for the derived factor group of Gn(w). Putting, for r ≦ 2, k ≦ 1,with subscripts reduced modulo n, we obtain the groups F(r, n, k) studied in (1) and (2), while the F(r, n, 1) are the ordinary Fibonacci groups F(r, n) of (3), (5) and (6). To conform with earlier notation, we write A(r, n, k) and A(r, n) for the derived factor groups of F(r, n, k), and F(r, n) respectively.

The subnormal structure of general linear groups

1986 ◽  
Vol 99 (3) ◽  
pp. 425-431 ◽  
Author(s):  
L. N. Vaserstein
Keyword(s):  
Normal Subgroup ◽  
Natural Number ◽  
Associative Ring ◽  
Inverse Image ◽  
General Linear ◽  
Linear Groups ◽  
Canonical Homomorphism ◽  
General Linear Groups ◽  
Subnormal Structure ◽  
Image Position

Let A be an associative ring with 1. For any natural number n, let GLnA denote the group of invertible n by n matrices over A, and let EnA be the subgroup generated by all elementary matrices ai, j, where aεA and 1 ≤ i ≡ j ≤ n. For any (two-sided) ideal B of A, let GLnB be the kernel of the canonical homomorphism GLnA→GLn(A/B) and Gn(A, B) the inverse image of the centre of GLn(A/B) (when n > 1, the centre consists of scalar matrices over the centre of the ring A/B). Let EnB denote the subgroup of GLnB generated by its elementary matrices, and let En(A, B) be the normal subgroup of EnA generated by EnB (when n > 2, the group GLn(A, B) is generated by matrices of the form ai, jbi, j(−a)i, j with aA, b in B, i ≡ j, see [7]). In particuler,is the centre of GLnA

scholarly journals Note on a theorem of Magnus

1969 ◽  
Vol 10 (3-4) ◽  
pp. 469-474 ◽  
Author(s):  
Norman Blackburn
Keyword(s):  
Normal Subgroup ◽  
Free Group ◽  
Reduction Theorem ◽  
The South ◽  
Cup Product ◽  
South West ◽  
Image Position ◽  
Independent Parameters

Magnus [4] proved the following theorem. Suppose that F is free group and that X is a basis of F. Let R be a normal subgroup of F and write G = F/R. Then there is a monomorphism of F/R′ in which ; here the tx are independent parameters permutable with all elements of G. Later investigations [1, 3] have shown what elements can appear in the south-west corner of these 2 × 2 matrices. In this form the theorem subsequently reappeared in proofs of the cup-product reduction theorem of Eilenberg and MacLane (cf. [7, 8]). In this note a direct group-theoretical proof of the theorems will be given.

Paradox of the class of all grounded classes

1953 ◽  
Vol 18 (2) ◽  
pp. 114-114 ◽  
Author(s):  
Shen Yuting
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Image Position ◽  
Special Case

A class A for which there is an infinite progression of classes A1, A2, … (not necessarily all distinct) such thatis said to be groundless. A class which is not groundless is said to be grounded. Let K be the class of all grounded classes.Let us assume that K is a groundless class. Then there is an infinite progression of classes A1, A2, … such thatSince A1 ϵ K, A1 is a grounded class; sinceA1 is also a groundless class. But this is impossible.Therefore K is a grounded class. Hence K ϵ K, and we haveTherefore K is also a groundless class.This paradox forms a sort of triplet with the paradox of the class of all non-circular classes and the paradox of the class of all classes which are not n-circular (n a given natural number). The last of the three includes as a special case the paradox of the class of all classes which are not members of themselves (n = 1).More exactly, a class A1 is circular if there exists some positive integer n and classes A2, A3, …, An such thatFor any given positive integer n, a class A1 is n-circular if there are classes A2, …, An, such thatQuite obviously, by arguments similar to the above, we get a paradox of the class of all non-circular classes and a paradox of the class of all classes which are not n-circular, for each positive integer n.

scholarly journals On the intersection of free subgroups in free products of groups

2008 ◽  
Vol 144 (3) ◽  
pp. 511-534 ◽  
Author(s):  
WARREN DICKS ◽  
S. V. IVANOV
Keyword(s):  
Free Product ◽  
Free Group ◽  
Additional Assumption ◽  
Free Products ◽  
Reduced Rank ◽  
Products Of Groups ◽  
Image Position ◽  
Free Subgroups ◽  
Special Case ◽  
Subgroup Theorem

AbstractLet (Gi | i ∈ I) be a family of groups, let F be a free group, and let $G = F \ast \mathop{\text{\Large $*$}}_{i\in I} G_i,$ the free product of F and all the Gi.Let $\mathcal{F}$ denote the set of all finitely generated subgroups H of G which have the property that, for each g ∈ G and each i ∈ I, $H \cap G_i^{g} = \{1\}.$ By the Kurosh Subgroup Theorem, every element of $\mathcal{F}$ is a free group. For each free group H, the reduced rank of H, denoted r(H), is defined as $\max \{\rank(H) -1, 0\} \in \naturals \cup \{\infty\} \subseteq [0,\infty].$ To avoid the vacuous case, we make the additional assumption that $\mathcal{F}$ contains a non-cyclic group, and we define We are interested in precise bounds for $\upp$. In the special case where I is empty, Hanna Neumann proved that $\upp$ ∈ [1,2], and conjectured that $\upp$ = 1; fifty years later, this interval has not been reduced.With the understanding that ∞/(∞ − 2) is 1, we define Generalizing Hanna Neumann's theorem we prove that $\upp \in [\fun, 2\fun]$, and, moreover, $\upp = 2\fun$ whenever G has 2-torsion. Since $\upp$ is finite, $\mathcal{F}$ is closed under finite intersections. Generalizing Hanna Neumann's conjecture, we conjecture that $\upp = \fun$ whenever G does not have 2-torsion.

Some Remarks on IA Automorphisms of Free Groups

1988 ◽  
Vol 40 (5) ◽  
pp. 1144-1155 ◽  
Author(s):  
J. McCool
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Free Group ◽  
Quotient Group ◽  
Free Groups ◽  
Finitely Generated ◽  
Generating Set ◽  
Automorphisms Of Free Groups ◽  
Identity Automorphism ◽  
Image Position

Let An be the automorphism group of the free group Fn of rank n, and let Kn be the normal subgroup of An consisting of those elements which induce the identity automorphism in the commutator quotient group . The group Kn has been called the group of IA automorphisms of Fn (see e.g. [1]). It was shown by Magnus [7] using earlier work of Nielsen [11] that Kn is finitely generated, with generating set the automorphismsandwhere x1, x2, …, xn, is a chosen basis of Fn.

Generators for certain normal subgroups of (2,3,7)

1965 ◽  
Vol 61 (2) ◽  
pp. 321-332 ◽  
Author(s):  
John Leech
Keyword(s):  
Normal Subgroup ◽  
Simple Group ◽  
Quotient Group ◽  
Hyperbolic Plane ◽  
Factor Group ◽  
Normal Subgroups ◽  
Finite Factor ◽  
Group 2 ◽  
Image Position ◽  
Symmetry Operations

The infinite groupis the group of direct symmetry operations of the tessellation {3,7} of the hyperbolic plane ((3), chapter 5). This has the smallest fundamental region of any such tessellation, and related to this property is the fact that the group (2, 3, 7) has a remarkable wealth of interesting finite factor groups, corresponding to the finite maps obtained by identifying the results of suitable translations in the hyperbolic plane. The simplest example of this is the group LF(2,7), which is Klein's simple group of order 168. I have studied this group in an earlier paper ((4)), showing inter alia that the group is obtained as a factor group of (2,3,7) by adjoining any one of the relationseach of which implies the others. The method used was to find a set of generators for the normal subgroup with quotient group LF(2,7) and, working entirely within this subgroup, to exhibit that any one of these relations implies its collapse. The technique of working with this subgroup had been developed earlier and applied in (6) to prove that the factor groupis finite and of order 10,752.

Symmetric powers, metabelian Lie powers and torsion in groups

1995 ◽  
Vol 118 (3) ◽  
pp. 449-466 ◽  
Author(s):  
Ralph Stöhr
Keyword(s):  
Finite Order ◽  
Free Group ◽  
Torsion Subgroup ◽  
Central Extensions ◽  
Torsion Element ◽  
Symmetric Powers ◽  
The One ◽  
Image Position ◽  
Special Case ◽  
Relatively Free Group

In this paper we study the homology of groups with coefficients in metabelian Lie powers, and apply the results to obtain information about elements of finite order in certain free central extensions of groups. Perhaps the most prominent example to which our results apply is the relatively free groupwhere Fd is the (absolutely) free group of rank d. Thus Fd(Bc) is the free group of rank d in the variety Bc of all groups which are both centre-by-(nilpotent of class ≤ c − 1)-by-abelian and soluble of derived length ≤ 3. It was pointed out in [1] that the order of any torsion element in Fd(Bc) divides c if c is odd and 2c if c is even. This, however, is a conditional result as it does not answer the question of whether or not there are any torsion elements in (1·1). Up to now, this question had only been answered in case when c is a prime number [1] or c = 4 [8]. In these cases Fd (Bc) is torsion-free if d ≤ 3, and elements of finite order do occur in Fd(Bc) if d ≥ 4. Moreover, the torsion elements in Fd(Bc) form a subgroup, and the precise structure of this torsion subgroup was exhibited in [1] in the case when c is a prime and in [8] for c = 4. In the present paper we add to this knowledge. On the one hand, we show that for any prime p dividing c the group Fd(Bc) has no elements of order p for all d up to a certain upper bound, which takes arbitrarily large values as c varies over all multiples of p. On the other hand, we show that for prime powers does contain elements of order p whenever d ≥ 4. Finally, we exhibit the precise structure of the p-torsion subgroup of when p ≠ 2. Precise statements are given below (Corollaries 1 and 2). Our results on (1·1) are a special case of more general results (Theorems 1′−3′) which refer to a much wider class of groups, and which are, in their turn, a consequence of our main results on the homology of metabelian Lie powers (Theorems 1–3).

Shelah's work on non-semi-proper iterations, II

2001 ◽  
Vol 66 (4) ◽  
pp. 1865-1883 ◽  
Author(s):  
Chaz Schlindwein
Keyword(s):  
Closed Subset ◽  
Stationary Subset ◽  
Finite Support ◽  
Countable Support ◽  
Final Model ◽  
Axiom A ◽  
Mahlo Cardinal ◽  
The One ◽  
Image Position ◽  
Special Case

One of the main goals in the theory of forcing iteration is to formulate preservation theorems for not collapsing ω1 which are as general as possible. This line leads from c.c.c. forcings using finite support iterations to Axiom A forcings and proper forcings using countable support iterations to semi-proper forcings using revised countable support iterations, and more recently, in work of Shelah, to yet more general classes of posets. In this paper we concentrate on a special case of the very general iteration theorem of Shelah from [5, chapter XV]. The class of posets handled by this theorem includes all semi-proper posets and also includes, among others, Namba forcing.In [5, chapter XV] Shelah shows that, roughly, revised countable support forcing iterations in which the constituent posets are either semi-proper or Namba forcing or P[W] (the forcing for collapsing a stationary co-stationary subset ofwith countable conditions) do not collapse ℵ1. The iteration must contain sufficiently many cardinal collapses, for example, Levy collapses. The most easily quotable combinatorial application is the consistency (relative to a Mahlo cardinal) of ZFC + CH fails + whenever A ∪ B = ω2 then one of A or B contains an uncountable sequentially closed subset. The iteration Shelah uses to construct this model is built using P[W] to “attack” potential counterexamples, Levy collapses to ensure that the cardinals collapsed by the various P[W]'s are sufficiently well separated, and Cohen forcings to ensure the failure of CH in the final model.In this paper we give details of the iteration theorem, but we do not address the combinatorial applications such as the one quoted above.These theorems from [5, chapter XV] are closely related to earlier work of Shelah [5, chapter XI], which dealt with iterated Namba and P[W] without allowing arbitrary semi-proper forcings to be included in the iteration. By allowing the inclusion of semi-proper forcings, [5, chapter XV] generalizes the conjunction of [5, Theorem XI.3.6] with [5, Conclusion XI.6.7].

scholarly journals Radially symmetric solutions of a class of singular elliptic equations

1990 ◽  
Vol 33 (2) ◽  
pp. 169-180 ◽  
Author(s):  
Juan A. Gatica ◽  
Gaston E. Hernandez ◽  
P. Waltman
Keyword(s):  
Boundary Value Problem ◽  
Elliptic Equations ◽  
Boundary Value ◽  
Open Unit ◽  
Open Unit Ball ◽  
Singular Elliptic Equations ◽  
Existence Of Positive Solutions ◽  
Radially Symmetric Solutions ◽  
Image Position ◽  
Special Case

The boundary value problemis studied with a view to obtaining the existence of positive solutions in C1([0, 1])∩C2((0, 1)). The function f is assumed to be singular in the second variable, with the singularity modeled after the special case f(x, y) = a(x)y−p, p>0.This boundary value problem arises in the search of positive radially symmetric solutions towhere Ω is the open unit ball in ℝN, centered at the origin, Γ is its boundary and |x| is the Euclidean norm of x.

A non-cyclic one-relator group all of whose finite quotients are cyclic

1969 ◽  
Vol 10 (3-4) ◽  
pp. 497-498 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Normal Subgroup ◽  
Residually Finite ◽  
Defining Relation ◽  
Finite Quotient ◽  
Finite Quotients ◽  
Image Position

Let G be a group on two generators a and b subject to the single defining relation a = [a, ab]: . As usual [x, y] = x−1y−1xy and xy = y−1xy if x and y are elements of a group. The object of this note is to show that every finite quotient of G is cyclic. This implies that every normal subgroup of G contains the derived group G′. But by Magnus' theory of groups with a single defining relation G′ ≠ 1 ([1], §4.4). So G is not residually finite. This underlines the fact that groups with a single defining relation need not be residually finite (cf. [2]).

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