Showing papers on "Free product published in 1973"

Finite and infinite cyclic extensions of free groups

Abstract: Using Stalling's characterization [11] of finitely generated (f. g.) groups with infinitely many ends, and subgroup theorems for generalized free products and HNN groups (see [9], [5], and [7]), we give (in Theorem 1) a n.a.s.c. for a f.g. group to be a finite extension of a free group. Specifically (using the terminology extension of and notation of [5]), a f.g. group G is a finite extension of a free group if and only if G is an HNN group where K is a tree product of a finite number of finite groups (the vertices of K), and each (associated) subgroup Li, Mi is a subgroup of a vertex of K.

New criteria for freeness in abelian groups

Abstract: A new criterion is established for an abelian group to be free. The criterion is in terms of an ascending chain of free subgroups and is dependent upon a new class of torsion-free groups. The result leads to

Free products of von Neumann algebras

Abstract: A new method of constructing factors of type III, called free product, is introduced. It is a generalization of the group cornstruction of factors of type IIl when the given group is a free product of two groups. If A1 and A2 are two von Neumann algebras with separating cyclic trace vectors and ortho-unitary bases, then the free product A1 * A2 of A 1 and A2 is a factor of type 11 without property r.

A Kurosh subgroup theorem for free pro--products of pro--groups

Abstract: Let C be a class of finite groups, closed under finite products, subgroups and homomorphic images. In this paper we define and study free pro-e- products of pro-e- groups indexed by a pointed topological space. Our main result is a structure theorem for open subgroups of such free products along the lines of the Kurosh subgroup theorem for abstract groups. As a consequence we obtain

Commutators as products of squares

Abstract: It is shown that if G is the free group of rank 2 freely generated by x and y, then xyx-ly-1 is never the product of two squares in G, although it is always the product of three squares in G. It is also shown that if G is the free group of rank n freely generated by x1, x2,I , xn, then xx2... xI is never the product of fewer than n squares in G. Let G be a group, G' its commutator subgroup, and G2 the fully invariant subgroup generated by the squares of the elements of G. It is well known that G2-D G'; indeed, if x, y are any element of G, then [x, y] = xyx-ly-1 = (Xy)2y-1X-1y)2(y-1)2 so that [x, y] is the product of three squares in G. The question naturally arises as to whether this number is minimal. That is, we ask if the equation (1) [x,y] = xyx-ly-y = a2b2 has solutions in G. In many cases the answer is in the affirmative; for example, if either x ory is of odd order or of order 2. In general, however, the answer is in the negative, and the purpose of this note is to prove the following: THEOREM 1. Let G={x, y} be the free group of rank 2 freely generated by x andy. Then the equation (1) has no solutions in G. Two proofs of this theorem will be given. The first (which is due to the second author) will be by representing G as a suitable transformation group and then showing the impossibility by matrix arguments. The second (which is due to the first author, who acted as referee for the original version of the paper) is both simpler and provides a proof of a related result, and will be by considering suitable homomorphic images of free groups of finite rank. No doubt a proof by a word argument alone is possible (in this connection see [3]) but seems difficult. Received by the editors June 9, 1972 and, in revised form, June 30, 1972. AMS (MOS) subject classifications (1970). Primary 20F05, 20F10, 20H05. I The first author wants to acknowledge support from the National Science Foundation. ? American Mathematical Society 1973 267 This content downloaded from 207.46.13.120 on Wed, 14 Sep 2016 05:43:36 UTC All use subject to http://about.jstor.org/terms 268 R. C. LYNDON AND MORRIS NEWMAN [July The result referred to above is the following: THEOREM 2. Let G be the free group of rank nfreely generated by x1, X2, x Xn. Then 1X2 *. x is never the product offewer than n squares in G. For the first proof of Theorem 1 we start with the classical modular group r = PSL(2, Z) = SL(2, Z)/{+I}. The elements of r will be written as matrices instead of cosets. We shall be careful to make plain when elements of SL(2, Z) are being discussed and when elements of r are being discussed. The group r is generated by and is the free product of the cyclic group {T} of order 2 and the cyclic group {ST} of order 3. The commutator subgroup rF of r is a free group of rank 2 and index 6, and is freely generated by

The free product of residually finite groups amalgamated along retracts is residually finite

Abstract: It is shown that residual finiteness is preserved by the generalized free product provided that the amalgamated subgroups are retracts of their respective factors. This result is applied to knot groups. The outcome is that the question of residual finiteness for knot groups need only be answered for prime knots.

Archimedean-like classes of lattice-ordered groups

Abstract: Suppose e denotes a class of totally ordered groups closed under taking subgroups and quotients by o-homomorphisms. We study the following classes: (1) Res (e), the class of all lattice-ordered groups which are subdirect products of groups in e; (2) Hyp (e), the class of lattice-ordered groups in Res (E) having all their 1-homomorphic images in Res (e); Para (e), the class of lattice-ordered groups having all their principal convex 1-subgroups in Res (e ). If C is the class of archimedean totally ordered groups then Para (C) is the class of archimedean lattice-ordered groups, Res(e) is the class of subdirect products of reals, and Hyp(e) consists of all the hyper archimedean lattice-ordered groups. We show that under an extra (mild) hypothesis, any given representable lattice-ordered group has a unique largest convex 1-subgroup in Hyp (e); this socalled hyper-E-kernel is a characteristic subgroup. We consider several examples, and investigate properties of the hyper-C-kernels. For any class e as above we show that the free lattice-ordered group on a set X in the variety generated by e is always in Res (e). We also prove that Res (C) has free products. Introduction. The theory presented in this paper grew out of an attempt to abstract the properties of the class of archimedean lattice-ordered groups (hence-the properties of the class of archimedean lattice-ordered groups (henceforth: 1-groups), and some of its subclasses. The classes we construct offer an appetizing alternative to studying varieties of 1-groups, one which quite surely has generalizations in other universal algebraic settings, although the reader may satisfy himself as to the special nature of many of the arguments. For any class C of totally ordered groups (henceforth: o-groups) which is closed under taking subgroups and quotients by o-homomorphisms, we construct Res (C), the class of i-groups which are subdirectly representable in a product of o-groups in C. An l1group is hyper-C if it is in Res (e) and every i-homomorphic image is in Res (C). The main result in ? 1 says that every representable i-group contains a unique maximal convex i-subgroup which is hyper-C; we call this the Presented to the Society, August 31, 1972; received by the editors April 3, 1972. AMS (MOS) subject classifications (1970). Primary 06A55.

The Computability of Group Constructions, Part I

Abstract: Publisher Summary This chapter discusses the computability of group constructions. The chapter also discusses relative Grzegorczyk hierarchy and free products with amalgamation. A free product with amalgamation is a useful construction when dealing with decision problems in groups because intuitively the normal form theorem yields decision procedures for such products modulo the decision procedures for the groups and the amalgamated subgroups. The chapter discusses strong Britton extensions.

On finitely generated subgroups which are of finite index in generalized free products

Abstract: Let G-=(A * B; U) be the free product of A and B with the subgroup U amalgamated. Various conditions are given which imply that every finitely generated subgroup H containing a (nontrivial) normal subgroup of G has finite index in G (in such a case we say G has the f.g.c.n. property). In particular, if A is a noncyclic free group and U is cyclic, then G has the f.g.c.n. property. We use this last result to give a combinatorial proof that Fuchsian groups have the f.g.c.n. property; this was first proved by Greenberg using non-Euclidean geometry.

A van Kampen theorem for separable algebras

Abstract: If the commutative ring S is the fibre product of R, and R2 over R, and if R2 maps onto R, then the fundamental group of S is the profinite free product of the fundamental groups of R1 and R2 with the fundamental group of R amalgamated. Van Kampen's theorem in algebraic topology asserts that, under suitable hypotheses, if

On the Frattini subgroups of certain generalized free products of groups

Abstract: Let G = (*I A )Hbe the generalized free product of the groups A, amalgamating the subgroup H We show that if G is residually finite and the groups A, have compatible H-filters then the Frattini subgroup (I'(G) is contained in the maximal G-normal subgroup in H If the groups A, are free and H is finitely generated of infinite index in one A, then N (G)=1 We also show that if H is simple then ?J (G)= 1 or H"' 1 The purpose of this paper is to continue our effort to answer the questions raised by Higman and Neumann [5]: whether the Frattini subgroup of a generalized free product can be larger than the amalgamated subgroup and whether such groups necessarily have maximal subgroups In [8] Whittemore answered these questions for generalized free products of finitely many free groups amalgamating a cyclic subgroup and for generalized free products of finitely many finitely generated abelian groups In [2] Djokovic and Tang showed that if G is the generalized free product amalgamating the subgroup H which satisfies the minimal condition on subgroups then the Frattini subgroup 4)(G) is contained in the maximal G-normal subgroup contained in H In this paper we shall extend the results in [2] and [8] In particular, we show that if G=(R* Ai),, is residually finite and the Ai have compatible H-filters then (I(G) is contained in the maximal G-normal subgroup contained in H In the case of G=(M1I Ai)H where each A, is free and H is finitely generated then d)(G)= 1 if H is of infinite index in one of the Ai We also show that if the amalgamated subgroup is simple then (D(G)= 1 or HG The notation used is in general standard: G=(A * B)II denotes the generalized free product of A and B amalgamating the subgroup H A subgroup N is said to be G-normal if N< G G\H denotes the set of elements of G not contained in H (a1, * * , a,) denotes the group generated by the elements a1, , an Received by the editors Febrwary 9, 1971 and, in revised form, February 18, 1972 AMS (MOS) subject classifications (1970) Primary 20E20, 20E30; Secondary 20F30

On magnus groups

Abstract: In this paper we consider groups of the form , where is a verbal subgroup of a normal divisor of a group , and is either free or the free product of certain groups. In the latter case we assume that is contained in the Cartesian subgroup. We prove that the factors of the lower central series of are torsion-free or even free abelian if the corresponding property is possessed by the factors of the lower central series of and .Bibliography: 7 items.

On the Word Problem and T-Fourth-Groups

Abstract: Publisher Summary This chapter discusses a word problem and T-fourth groups. The chapter also discusses one-fourth condition, triangle condition, and preliminary lemmas and presents main results. The chapter describes a theorem where K is the free product of T-fourth groups with an infinite cyclic group amalgamated. K has a solvable word problem.

Subgroups of free products with amalgamated subgroups: A topological approach

Abstract: The structure of an arbitrary subgroup of the limit of a group system is shown to be itself the limit of a group system, the elements of which can be described in terms of subgroups of the original group system. Introduction. It is the intent of this paper to describe the structure of an arbitrary subgroup of the limit of a group system in terms of subgroups of the members of that system. The description that is obtained is given by the following theorem: Theorem. Let H be any subgroup of the limit of the group system IGa, H as' ' as: a> s e A^' Then H is itself the limit of a group system. Each of the factors of H is the free product of a conjugate of a subgroup of one of the G and a free group (which may be trivial). Each of the amalgamating groups is the free product of a conjugate of a subgroup of one of the H ~ and a free group (which may be trivial). The general approach to this problem will be topological. That is to say, a complex K whose fundamental group is the limit of the given group system and whose topological composition parallels the algebraic composition of that group system will be constructed. The structure of a subgroup H may then be determined by computations of the fundamental group of a covering space of K. §1 will set the notation and terminology that will be used throughout, and §11 will give the construction of the model complex K with some properties of its covering spaces. §111 will give the proof of the main theorem, and will discuss the relation of the approach taken here to results previously obtained in similar settings. Received by the editors September 23, 1971 and, in revised form, November 22, 1972. AMS (MOS) subject classifications (1970). Primary 20E30, 20F15; Secondary 55A05, 55A10. Key ivords and phrases. Group system, fundamental group, covering space, tree product, HNN group, generalized free product. (1) Part of this material was included in the author's Ph.D. dissertation submitted to Dartmouth College and supervised by Professor Edward M. Brown. Copyright © 1973, American Mathematical Society

Solvable Groups, Automorphism Groups, and Representation Theory*

Abstract: Publisher Summary This chapter proves the theorem that if a solvable group G admits a nilpotent free fixed point free group of operators A (i.e. C G (A) = 1) where (|A|, |G|) = 1 then the Fitting height of G bounded above by the composition length of A.