Certain Classes Of Groups Research Articles

Flat Riemannian manifolds are boundaries

In this paper we prove that each compact flat Riemannian manifold is the boundary of a compact manifold. Our method of proof is to construct a smooth action of (ℤ2) k on the flat manifold. We are independently preceded in this approach by Marc W. Gordon who proved the flat Riemannian manifolds, whose holonomy groups are of a certain class of groups, bound. By analyzing the fixed point data of this group action we get the complete result. As corollaries to the main theorem it follows that those compact flat Riemannian manifolds which are oriented bound oriented manifolds; and, if we have an involution on a “homotopy flat” manifold, then the manifold together with the involution bounds. We also give an example of a nonbounding manifold which is finitely covered byS 3 ×S 3 ×S 3.

Algebraic Fiber Bundles

When $X$ is a finite simplicial complex and $G$ is any of a certain class of groups, a classification of $G$-principal bundles over $X$ in terms of projective modules over a ring $R(G,X)$ is given. This generalizes Swan’s classification of vector bundles and uses the results of Mulvey. Often, $R$ can be taken to be noetherian; in this case ${\text {Spec}}(R)$ is usually reducible with "cohomologically trivial" irreducible components. Information is derived concerning the nature of projective modules over such rings, and some results are obtained indicating how such information reflects information about $X$.

Algebraic fiber bundles

When X X is a finite simplicial complex and G G is any of a certain class of groups, a classification of G G -principal bundles over X X in terms of projective modules over a ring R ( G , X ) R(G,X) is given. This generalizes Swan’s classification of vector bundles and uses the results of Mulvey. Often, R R can be taken to be noetherian; in this case Spec ( R ) {\text {Spec}}(R) is usually reducible with "cohomologically trivial" irreducible components. Information is derived concerning the nature of projective modules over such rings, and some results are obtained indicating how such information reflects information about X X .

Conjugate purity and infinite groups

AbstractIn Balogun (1974), we proved that a finite group in which every subgroup is conjugately pure is necessarily Abelian and we left open the infinite case. In this paper we settle this problem positively for soluble, locally soluble groups and certain classes of groups which include the FC-groups. In the last section of this paper we characterize groups which are conjugately pure in every containing group.Subject classification (Amer. Math. Soc. (MOS) 1970): 20 E 99.

STRONGLY PRIME GROUP RINGS

Abstract A ring R is (right) strongly prime (SP) if every nonzero two sided ideal contains a finite set whose right annihilator is zero, SP rings have been studied by Handelman and Lawrence who raised the problem of characterizing SP group algebras. They showed that if R is SP and G is torsion free Abelian, then the group ring RG is SP. The aim of this note is to determine some more group rings which are SP. For a ring R we also define the strongly prime radical s(R). We then show that s(R)G = s(W) for certain classes of groups.

A fusion lemma for a certain class of groups of characteristic 2-type

A fusion lemma for a certain class of groups of characteristic 2-type

The verbal topology of a group

We define for any group G a topology on G called the verbal topology of G and study groups which satisfy the minimal condition on the closed sets in the verbal topology, called min-cbsed. The main results of the paper establish with easy proofs that certain classes of groups satisfy min-closed. Since it is shown that groups with min-closed are C&groups (see [2] or [5]), these results give some new examples of CZ-groups: for example, finitely generated Abelian-bynilpotent groups. Also, since centralizers are closed in the verbal topology, another consequence is an easy proof of the fact that finitely generated Abelianby-nilpotent groups satisfy the minimal condition on centralizers (or, equivalently, the maximal condition on centralizers). This was proved in a stronger form, but with a rather difficult proof, by Lennox and Roseblade [3], answering a question of P. Hall. The investigation which led to the results mentioned above began with a search for discriminating groups (see Neumann [4]). In this connection we prove that every group with min-closed has a discriminating subgroup of finite index.

Classes of groups defined by series or normal factor coverings

Of primary interest in this work is the establishing of new closure properties for certain classes of groups defined by means of series or normal factor coverings. (Here, series is as defined by Robinson [IO, p. 91, and norma factor covering is as defined by Durbin [3].) We begin by identifying cxactl!; those classes of groups with which we shall deal. After the identification of these classes, we discuss the new closure properties. Let r’. be a variety of groups defined by a set of words CIT. Also, let pW be the subgroup theoretical property IV-margina [IO, p. 91. Then the series classes with which we deal are @V, P,V and (PI&‘)‘), , the latter being defined by G E ( pW)s if and only if G has ap W-series. (The operations P, fi, and the notion of a pI&‘-series are defined by Robinson [ 10, pp. 12-131.) These classes receive attention in [I I] and are characterized in an elegant fashion by Hickin and Phillips in a recent work [5]. 1% :e note that some familiar classes result whenever W consists of the commutator word. For then ‘Y is the v-ariet!of all abelian groups, and the classes of SN-groups, S-groups, Z-groups (in the sense of Kurosh [6]) are given by i%-, @,,-l ‘, ( pW)s , respectively. Now let us define some classes of groups by means of normal factor coverings. Define an operation F, by G t F,X if and only if G has a normal factor covering with X-factors, where 3 is any class of groups. If I. is a variety of groups defined bv a set of words IX, then in the sequel we shall be concerned with the normai factor covering classes F%Y“ and (pW), , which is defined by G E ( pW), provided G has a normal factor covering {rC, , H, : a: E A] such that H%/K,, is W-marginal in G/K, , for each a E A. Whenever W consists of the commutator word, we mention that the classes F,‘Y . and (pW), have appeared in the literature [I, 2, 31.

Infinitary model theory of abelian groups

The paper is a survey of results in the model theory of abelian groups, dealing with two sorts of problems: finding invariants which classify groups up toL λκ-equivalence; and determining whether certain classes of groups are definable inL λκ.

Recurrent random walks on certain classes of groups

This article is divided into two parts: in the first we give some results about renewal and normality of a recurrent random walk (r.w.) on an abelian group, without the Harris hypothesis, which will extend the theorems of S.C. Port and C.J. Stone ([8]) to a larger class of functions. They are stated in the Theorems 1.14 and 1.15. The technique will be to approximate the recurrent r.w. by a Harris recurrent r.w., for which the recent results of A. Brunel and D. Revuz ([2–4]) hold.

Central approximate units in a certain ideal of 𝐿¹(𝐺)

In this paper we show that for a locally compact group G G the ideal L 0 ( G ) = { f | f ∈ L 1 ( G ) , ∫ f = 0 } {L^0}(G) = \{ f|f \in {L^1}(G),\smallint f = 0\} of L 1 ( G ) {L^1}(G) has multiple approximate units belonging to the center of L 0 ( G ) {L^0}(G) iff G G has a basis of invariant neighbourhoods of 1 and if all conjugacy classes of G G are precompact, or, equivalently, the group of inner automorphisms is precompact in the group of all topological automorphisms. In a sense this is part of the problem to characterize certain classes of groups by properties of the group algebra.

A question of Babai on groups

László Babai raised the question whether every infinite group G contains a set B of elements, of cardinal equal to the order of G, such that for elements a, b, c of B the equation a = bc−1b implies a = c. We show that the answer is affirmative for certain classes of groups, including the soluble groups and the countable groups, if the Axiom of Choice is assumed, but negative, even for abelian groups, if a different axiom, compatible with Zermelo-Fraenkel set theory but incompatible with the Axiom of Choice, is assumed.

Semiisomorphisms of certain classes of groups

Semiisomorphisms of certain classes of groups

Radical Classes need not have a Unique Maximal R0-Closed Subclass

It was shown in [1] that certain classes of groups which are closed under quotients, and extensions contain a unique maximal inclosed subclass. These results prompted the question whether there exists a class of groups which is closed under quotients and extensions and yet does not have a unique maximal R0-closed subclass. This note provides an example of such a class.

Stable automorphisms of certain classes of groups

Stable automorphisms of certain classes of groups

The word and conjugacy problems for the knot group of any tame, prime, alternating knot

The decision problems in the title are solved using the solutions of the word and conjugacy problems obtained by Lyndon and Schupp, respectively, for certain classes of groups.

Inclusion problem for a certain class of groups

Inclusion problem for a certain class of groups

Symmetry adapted functions belonging to the crystallographic groups

Various authors have discussed methods of obtaining the irreducible representations of groups containing nontrivial normal subgroups, in terms of the representations of the normal subgroup and its factor group. These results have applied to special cases, such as to normal subgroups of prime index or to the case in which a subgroup exists isomorphic to the factor group and having one element in each coset of the normal subgroup. These results have been extended to a general theory, which not only includes all the results of the papers cited, but also enables one to obtain very easily the characters and representations of certain classes of groups heret ofore obtainable only by special methods. Examples of such groups are the crystallographic “double” groups which have been discussed by Bethe, Elliott, and Opechowski as well as those crystal-lographic lattice groups for which there is no subgroup isomorphic to the corresponding point group. The basis of this theory is a canonical form for the representations of a semidirect product of two groups; a generalization of the representation theory of direct products which is developed in the present paper.

On the Indecomposable Representations of a Certain Class of Groups

On the Indecomposable Representations of a Certain Class of Groups

On Some Types of Topological Groups

As is well-known the so-called fifth problem of Hilbert on continuous groups was solved by J. v. Neumann [14]2 for compact groups and by L. Pontrjagin [15] for abelian groups. More recently, it is reported, C. Chevalley [6] solved it for solvable groups.3 Now it seems, as H. Freudenthal [7] clarified for maximally almost periodic groups, that the essential source of the proof of Hilbert's problem for these groups lies in the fact that such groups can be approximated by Lie groups. Here we say that a locally compact group G can be approximated by Lie groups, if G contains a system of normal subgroups {Na} such that G/Na are Lie groups and that the intersection of all Na coincides with the identity e. For the brevity we call such a group a group of type (L) or an (L)-group. In the present paper we shall study the structure of such (L)-groups, and apply the result to solve the Hilbert's problem for a certain class of groups, which contains both compact and solvable groups as special cases. We shall be able to characterize a Lie group G, for which the factor group GIN of G modulo its radical N is compact, completely by its structure as a topological group. The outline of the paper is as follows. In ?1 we study the topological structure of the group of automorphisms of a compact group and prove theorems concerning compact normal subgroups of a connected topological group, which are to be used repeatedly in succeeding sections. In ?2 come some preliminary considerations on solvable groups, whereas finer structural theorems on these groups are, as special cases of (L)-groups, given later. In ?3 we prove some theorems on Lie groups. The theorems here stated are not all new, but we give them here for the sake of completeness, and thereby refine and modify these theorems so as to be applied appropriately in succeeding sections.4 After these preparations we study in ?4 the structure of (L)-groups. In particular, it is shown 'that the study of the local structure and the global topological structure