If C is a class of groups, we denote by RC the class of groups which are residually in C i.e. G ∈ RC if and only if 1 ≠ g ∈ G implies that there exists a normal subgroup N of G such that g ∈ N and G/N ∈ C. A group G is residually a finite p-group if it belongs RFp, where Fp denotes the class of finite p-groups. One also says that the groups in RFp are residually of order equal to a power of the prime p. Given a group G with one defining relator r, one might ask for conditions on the “form” of the relator that would guarantee that G have certain residual properties. In this context, Baumslag (1971) has proved that if all the exponents of the generators appearing in r are positive, then G is residually solvable. In the same paper he also concerned himself with the residual nilpotence of one-relator groups, and found that the situation there was much more complicated. If one goes one step further and asks for conditions that will ensure that for a given prime p the one-relator group be residually a finite p-group, then very little seems to be known. Of course, if one takes r to be one of the generators: then G is freely generated by the remaining generators, and hence is in RFp for all primes p (Mahec (1949), Lazard (1965), 3.1.4). Our main purpose in this paper is to develop methods of generating examples of one-relator groups that are residually of order equal to a given prime p.

The purpose of this note is to characterize maximal right ideals of T0(G), the (left) mear-ring of transformations from a group (G, +) into itself which leave zero fixed. Using this characterization, we answer some questions raised by Heatherly (1972).

The full transformation semigroup F(X) is extremely important. As far back as 1952, Malcev (1952) showed that its automorphism group consisted of mappings of the form g-1 · g, g ∈ G(X). The present work is devoted to a generalization of this result.

In this paper we discuss the existence of an are of minimal length joining two arbitrary, yet fixed points in a complete metric space, where the metric is restricted only by the properties (A) and (B) given below. It is shown that under these conditions an arc of least length joining any two fixed points exists, and is unique. In addition, its length is shown to be equal to the metrie distance between the points.

In the study of locally finite varieties of groups it has often been illuminating to consider when a group A is a factor (i.e. quotient of a subgroup) of a group B. We write A ⋨ B to express this and say that A is involved in B. It follows from elementary isomorphism theorems that the relation ⋨ is a partial order on any set of finite groups. The conjecture that we consider in this paper (and to which we only give the beginning of an answer) is the following:

Our main purpose in this paper is to obtain more precise information about two problems which we investigated in Hartley (1971a). They are as follows: Problem 1. Let G be a countable locally finite group and π be a set of primes. Suppose that G = HK, H◃G, H∩K = 1, where H is a normal π′-subgroup of G, K is a π-group and Ck(H) = 1. If we assume that the Sylow (that is, maximal) π-subgroups of G are conjugate, what can we say about the structure of K?

It is well-known that for any set X, px, the semigroup of all partial transformations on X, can be embedded in Jx∪a for some a ∉ X (see for example Clifford and Preston (1967) and Ljapin (1963)). Recently Magill (1967) has considered a special case of what we call ‘generalised partial transformation semigroups’. We show here that any such semigroup can always be embedded in a full transformation smigroup in which the operation is not in general equal to the usual composition of mappinas. We then examine conditions under which such a semigroup, (J x, θ), is isomorphic to the semigroup, under composition, of all transformations on the same set X.

Let R be an integral domain. It is well known (see Lambek (1971), Stenström (1971)), that idempotent filters of right ideals, torsion radicals and trosio theories are in one-to-one correspondence, but that different idempotent filters F of right ideals may lead to the same rings of quotiens Rf. We have always R ⊃ Rf ⊂Qmax(R). Given this situation one can ask a number of questions. For example: Describe all different idempotent filters for a given ring. Determine all different rings of quotients. When do different filters lead to the same ring of quotients? When are all rings between R and Qmax(R) of the form RF. When is every RF of the form RS−1, where S is an Ore system?

In recent years, there has been considerable interest in the refining of thin plate theories. In this paper, the method of matching asymptotic expansions is used to obtain one such refinement which is believed to be an improvement on several previous results. Previous authors(Habip (1967), Widera (1969)) attempted such refinemnts within the framework of a partially nonlinear theory of elasticity whereas in the present work all terms neglected by these authors have been retained.

Let be regular and unvivalent in the unit disk D and map D onto a region R. A point w0∈ R is called a star-center point of f(z), or of R, if tf(z)+(1−t)w0∈ R for z∈D and 0≦ t≦ 1. In this paper we consider only functions of the form (1) where w = 0 is a star-center point, i.e., those functions that are starlike with respect to the origin.