An enumeration of known simple groups has been given by Dickson [17]; to this list, he made certain additions in later papers [15], [16]. However, with but five exceptions, all known simple groups fall into infinite families; these five unusual simple groups were discovered by Mathieu [21], [22] and, after occasioning some discussion [20], [23], [27], were relegated to the position, which they still hold, of freakish groups without known relatives. Further interest is attached to these Mathieu groups in virtue of their providing the only known examples (other than the trivial examples of the symmetric and alternating groups) of quadruply and quintuply transitive permutation groups.

A single valued function D(x, y) is a metric for a topological space provided that for points x, y, z of the space:

1. D(x, y) ≽ 0, the equality holding if and only if x = y,

2. D(x, y) = D(y, x) (symmetry),

3. D(x, y) + D(y, z) ≽ D(x, z) (triangle inequality),

4. x belongs to the closure of the set M if and only if D(x, m) (m element of M) is not bounded from 0 (preserves limit points).

With each set of points S of a distance space there is associated a set of non-negative real munbers D(S) called the distance set of 5. The number x is an element of D(S) if and only if x is a distance between some pair of points of 5. The number zero is always an element of any distance set and no two distinct elements are equal.

Among the classical field theories, general relativity theory occupies a somewhat peculiar place. Unlike those of most other field theories, the field equations in relativity theory are non-linear. This implies that many facts, well known in linear theories, have no analogues in general relativity theory, and conversely. The equations of motion of the sources of the gravitational field are contained in the field equations, a fact which does not apply for the motion of an electron in the electromagnetic field. Conversely, it is difficult to define the notion of a wave (familiar in electrodynamics) in relativity theory; for, the linear principle of superposition is crucial for the existence of waves (at least in the sense that the notion of a wave is normally used).

The purpose of the present work is to give, as a continuation of the writer's study of Galois theory for general rings ([8], [9], [10]), a kind of Galois theory for general, non-commutative and non-semisimple rings, which includes, at least in its main features, the Kaloujnine-Jacobson Galois theory of non-normal fields ([3]; cf. [4], [5]). To deal with the non-commutativity we bring to the fore certain double-moduli rather than self-composites, while the non-semisimplicity is manipulated by the method and idea used in the writer's above mentioned study on (normal) Galois theory and commuter systems of nonsemisimple rings. (For the normal Galois theory of rings cf. [1], [2], [6], [7], [11], besides the above.) Some of our arguments may even serve to make some simplification in Jacobson's treatment of ordinary fields.

Let X be a topological space, that is, a space with open sets such that the union of any collection of open sets is open and the intersection of any finite number of open sets is open. A covering of X is a collection of open sets whose union is X. The covering is called countable if it consists of a countable collection of open sets or finite if it consists of a finite collection of open sets ; it is called locally finite if every point of X is contained in some open set which meets only a finite number of sets of the covering. A covering is called a refinement of a covering U if every open set of X is contained in some open set of . The space X is called countably paracompact if every countable covering has a locally finite refinement.

This paper is a result of an investigation into general methods of determining the irreducible characters of GL(n, q), the group of all non-singular linear substitutions with marks in GF(q), and of the related groups, SL(n, q), PGL(n, q), PSL(n, q), the corresponding group of determinant unity, projective group, projective group of determinant unity, respectively.

1.1. This paper is a continuation of the papers of the author [14], [15]. We begin by recapitulating the main definitions. If {n,} is an increasing sequence of positive integers, the value of the characteristic or the counting function ω(n) of {nv} is, for any , the number of n satisfying the inequality . Suppose that A is a linear method of summation corresponding to the transformation

In two papers I have proved that there are only a finite number of quadratic fields [6] and of cyclic cubic fields [7] in which Euclid's algorithm (E.A.) holds. Davenport has shown by a different method that there are only a finite number of quadratic fields [1, 2], of non-totally real cubic fields [3, 4] and of totally complex quartic fields in which E.A. holds.

The definition of a regular polyhedron may be enunciated as follows:

(α) A polyhedron is said to be regular if its faces are equal regular polygons, and its vertex figures are equal regular polygons

In a recent note1 I gave three examples of uniform non-regular polyhedra, which I called facially-regular, using the definition:

(β) A polyhedron is said to be facially-regular if it is uniform and all its faces are equal.

A collection of n + 1 convex subsets of a Euclidean space E will be called an n-set in E provided each n of the sets have a common interior point although the intersection of all n + 1 interiors is empty. It is well-known that if {C0,C1} is a 1-set, then C0 and C1 can be separated by a hyperplane.

The present paper constitutes a continuation of the ideas and methods of M. Morse and M. Heins [1]. As in that work the subject treated is the theory of deformation classes of meromorphic functions and interior transformations. There the functions considered were defined over the open disc < 1 and had only a finite number of zeros, poles and branch point antecedents. It is possible to transfer the results obtained to the situation where the domain of definition is any simply-connected domain of hyperbolic type or, alternatively, of parabolic type.

The object of this note is to prove the following theorem.

THEOREM. Let A be a division ring with centre Z, and suppose that for every x in A, some power (depending on x) is in . Then A is commutative.

Let R be a semi-ordered linear space, that is, a vector lattice in Birkhoff's terminology [2]. An element a ∈ R is said to be discrete, if for every element x ∈ R such that there exists a real number a for which x = aa. For every pair of discrete elements a, b ∈ R we have or there exists a real number a for which b = aa or a = ab.

A configuration is a system of m points and n lines such that each point lies on μ of the lines and each line contains v of the points. It is usually denoted by the symbol (mμ,nμ) with mμ = nv. Two configurations corresponding to the same symbol are said to be equivalent if there exist 1-1 mappings of the points and lines of one onto the points and lines of the other which preserve the incidence relations.

In a recent paper [5], general methods were described for the dissection of a square into a finite number n of unequal non-overlapping squares. In this note, examples of such perfect squares are given in which the sides and elements are relatively small integers; in particular, a dissection of a square into 24 different elements, which is believed to be the squaring of least order known at the present time.

The theory of modular representations of the symmetric group was studied first by Nakayama (5, 6), and later by Thrall and Nesbitt (11) and Robinson (7, 8, 9). Nakayama built up his elaborate theory of hooks for the express purpose of studying this problem, while Robinson's extensive work on the various phases of the relationship between Young diagrams, skew diagrams and star diagrams on the one hand, and representations of the symmetric group on the other, culminating in a set of relations among the degrees of the representations, serves as a starting point for this paper.

Although the title of the paper suggests that the nature of the problem considered is group theoretic, our methods are almost completely combinatorial and number theoretic in nature, the group theory entering only insofar as it leads us to various recursions that we study. Let Tn denote the number of solutions of x2 = 1 in Sn the symmetric group of degree n.

Let p(x) > 0, q(x) be two real-valued continuous functions on . Suppose that the differential equation with the real parameter λ

Les systémes différentiels dont il est question dans ce travail, sont du type suivant: