Banach (1, pp. 242-243) defines, for two Banach spaces X and Y, a number (X, Y) = inf (log (‖L‖ ‖L-1‖)), where the infimum is taken over all isomorphisms L of X onto F. He says that the spaces X and Y are nearly isometric if (X, Y) = 0 and asks whether the concepts of near isometry and isometry are the same; in particular, whether the spaces c and c0, which are not isometric, are nearly isometric. In a recent paper (2) Michael Cambern shows not only that c and c0 are not nearly isometric but obtains the elegant result that for the class of Banach spaces of continuous functions vanishing at infinity on a first countable locally compact Hausdorff space, the notions of isometry and near isometry coincide.

This paper presents a list of all finite groups having exactly six and seven conjugate classes and an outline of the background necessary for the proof, and gives, in particular, two results which may be of independent interest. In 1903 E. Landau (8) proved, by induction, that for each the equation

*

has only finitely many solutions over the positive integers.

An involution * of a ring A is a one-one additive mapping of A onto itself such that (xy)* = y*x* and x** = x for all x, y ∊ A. If A is an algebra over a field Φ, one makes the additional requirement that (λx)* = λx* for all λ ∊ Φ, x ∊ A. S will generally denote the set of symmetric elements s* = s, K the set of skew elements , and Z the centre of A.

All rings considered are associative. We show that if a homomorphically closed class P1 of rings is hereditary in the sense that every ideal of a ring in P1 is also in P1, then the lower Kurosh radical construction terminates at P3. This is an improvement on the result of Anderson, Divinsky, and Sulinski (3) showing that the lower radical construction terminates at P2 provided P1 is homomorphically closed, hereditary, and contains all zero rings. Examples are given to show that the third step is actually attained in some constructions.

It is now known (3) that if is a regular rank ring, then the rank function can be extended to the matrix ring in such a way that R(a) = R(a ꕕ n) ; here, a is an arbitrary element of is the n × n diagonal matrix with a for each entry on the diagonal, and R denotes rank in and also in . It is also known (2) that every regular rank ring has a rankmetric completion which is again a regular rank ring.

Let PGL(2, ƒ) denote the group of all Moebius transformations

over a field F. The object of this paper is to prove the following theorem.

Theorem 1. G is an S2-group and the centre of G is trivial if and only if G is isomorphic to a group PGL(2, ƒ), char ƒ ≠ 2.

This theorem was proved for finite groups in (1). The present paper extends the result to infinite groups and also improves the method of proof used in that paper. Many of the theorems given there were proved for infinite groups and are used here with appropriate references.

In this paper we shall be concerned with the following problem: If h is an elliptic function and h(z) = ƒ(g(z)), what can be said about the functions ƒ and g? In order to simplify the discussion we introduce some basic definitions.

Definition 1. A meromorphic function h(z) = ƒ(g(z)) is said to have ƒ(z) and g(z) as left and right factors, respectively, provided that either ƒ(z) is non-linear and meromorphic and g(z) is non-linear and entire or f(z) is rational and g(z) is meromorphic.

Given an algebra A, the elements of A induce linear operators on A by left and right multiplication. Various authors have studied Banach algebras A with the property that some or all of these multiplication maps are completely continuous operators on A ; see (1-5). In (3)1. Kaplansky defined an element u of a Banach algebra A to be completely continuous if the maps a ⟶ ua and a ⟶ au, a ∊ A, are completely continuous linear operators.

If K and L are chain complexes of abelian groups (to which we restrict ourselves throughout this paper), then denotes the graded hyperhomology group of K and L, as defined in Car tan and Eilenberg (1) by means of free double complex resolutions of K and L. Hyperhomology groups have proved convenient in proving various versions of the Künneth theorem (see, for example, (4; 1; 2)).

Let ƒ be a map from a topological space X into a topological space F. We say that ƒ is proper in case ƒ is closed continuous and ƒ-1(y) is compact for all y ∊ Y. Proper maps have been extensively studied, see for example (3, Chapter I, §10) or (6). (The definition of a proper map given above is different from, but equivalent to, that given by Bourbaki in (3). In (6) only surjective proper maps are considered and these maps are called ƒitting maps.) It is known that if ƒ is a proper map, then X is compact if and only if ƒ(X) is compact, and X is paracompact if and only if ƒ(X) is paracompact.

We shall use the term n-configuration for the complete graph having n vertices, which we visualize as a regular convex polygon. The sides and diagonals will be referred to as edges. A complete sub-configuration of p vertices (subset of p vertices with all interconnecting edges) will be called a p-tuple. A 3-tuple will be called a triple or triangle, a 4-tuple a quadruple, etc.

Let n be an ordinal. A partial ordering P of the ordinals T = T(n) = {w: w < n} is called natural if x P y implies x ⩽ y.

A natural partial ordering, hereafter abbreviated NPO, of T(n) is thus a coarsening of the natural total ordering of the ordinals. Every partial ordering of a finite set 5 is isomorphic to a natural partial ordering. This is a consequence of the theorem of Szpielrajn (5) which states that every partial ordering of a set may be refined to a total ordering. In this paper we consider only natural partial orderings. In the first section we obtain theorems about the lattice of all NPO's of T(n).

Let g be a semisimple Lie algebra over an algebraically closed field K of characteristic 0. Let h be a Cartan subalgebra of g and let Δ be the root system of g with respect to h.

It is well known that given the polynomial algebra (for definitions, see §2), an algebra of type τ, and a sequence a of elements of , one can define a congruence relation θa of such that the factor algebra is isomorphic to the subalgebra of generated by a, and the isomorphism is given in a very simple way.

By a PQ (product-quotients) algebra, we mean a non-empty set together with three single-valued and not necessarily associative operations ., / , \ that we shall treat as product, right quotient, and left quotient although we require no relation between them. The theory of binary systems provides the following examples:

A. is the set of all subsets of a groupoid G (which may be a semigroup) with the operations defined by:

The present investigation is based on two papers: “On the determination of numbers by their sums of a fixed order,” by J. L. Self ridge and E. G. Straus (4), and “On the determination of sets by the sets of sums of a certain order,” by B. Gordon, A. S. Fraenkel, and E. G. Straus (2).

First of all, we explain the terms implicit in the above titles. Throughout these considerations we use the term “set” to mean “a totality having possible multiplicities,” so that two sets will be counted as equal if, and only if, they have the same elements with identical multiplicities. In the most general sense the term “numbers” of (4) can be replaced by “elements of any given torsioniree Abelian group.”

Le but de cet article est d'établir une théorie des catégories additives axée sur l'idée de structure. Etant donné ce que nous connaissons aujourd'hui des catégories additives, il était tout naturel qu'une telle étude fut entreprise, et ce, dans l'optique d'une généralisation de la théorie classique (Wedderburn-Artin-Jacobson) des anneaux non-commutatifs avec élément unité. C'est dans “On the radical of a category” de G. M. Kelly (2) que l'on retrouve une amorce à l'étude des catégories additives, telle que nous la concevons dans cet article.