The characters of the representations of a finite group G over a field K of characteristic zero generate a ring oK(G) of functions on G, the K-character ring of G, which is readily seen to be Zϕ1 + . . . + Zϕn, where Z is the ring of rational integers and ϕ1, . . . , ϕn are the characters of the different irreducible representations of G over K. The theorem that every irreducible representation of G over an algebraically closed field Ω of characteristic zero is equivalent to a representation of G over the subfield of Ω which is generated by the g0th roots of unity (g0 the exponent of G) was proved by Brauer (4) via the theorems that

(1) OΩ(G) is additively generated by the induced characters of representations of elementary subgroups of G, and

(2) the irreducible representations over 12 of any elementary group are induced by one-dimensional subgroup representations (3).

The problem of developing an abstract integration theory has been approached from many angles (6). The most general of several definitions based on the norm topology is that of Birkhoff (1), which includes the well-known and widely used Bochner integral (3).

The original Birkhoff formulation was based on the notion of unconditional convergence of an infinite series of elements in a Banach space and the closed convex extensions of certain approximating sums.

The simplest example of two non-isomorphic groups with the same character tables is provided by the non-abelian groups of order p3, p ≠ 2. Let G1 be the one of exponent p and let G2 be the other. If Q denotes the field of rational numbers, then Berman (2) has shown that QG1 ≈ QG2, where QGi denotes the rational group algebra. In this note we shall show that the corresponding statement is false for ZGi where Z is the ring of rational integers. More explicitly we shall show that ZG1 does not contain a unit of order p2 so that it is impossible to embed ZG2 in ZG1.

Let G be a finite group and R a Dedekind domain with quotient field K. We denote by RG the group ring of formal linear combinations of elements of G with coefficients in R. By an RG-module we understand a unital left RG-module which is finitely generated and torsion-free as R-module. In particular, if R is a principal ideal domain this is equivalent to considering representations of G by matrices with entries in R.

Erdélyi and Sneddon (4) have reduced the dual integral equations (4, (1.4))

where Ψ is unknown, to a single Fredholm integral equation (4, (4.4)), from the solution of which Ψ is explicitly obtainable. Their work extended and clarified an investigation by Cooke (1), placing it in a context of standard integral transforms. Cooke's reduction was obtained after consideration of the Fredholm integral equation obtained by Love (8) in discussing Nicholson's problem of the electrostatic field of two equal circular coaxial conducting disks (9).

Hall's algebra (3) is an algebra over the field V(p) of rational functions in the indeterminate p with coefficients in the field V of complex numbers. The basis of the algebra consists of elements Gλ which are in one-one correspondence with the set of all partitions (λ) and whose multiplication "constants" are the "Hall polynomials" , i-e.

Let R be the discrete group ring of the group G over the ring A. In this paper we attempt to find necessary and sufficient conditions on G and A so that R will have some standard ring-theoretic property ; among the properties considered are those of being artinian, regular, self-injective, and semi-prime.

The contents of this paper form essentially the author's doctoral thesis. The author would like to thank his supervisor Dr. J. Lambek for his generous encouragement and continued interest.

A partially balanced incomplete block (PBIB) design with m-associate classes is defined by Bose and Shimamoto (4) as follows:

(i) The experimental material is divided into b blocks of k units each, different treatments being applied to the units in the same block.

(ii) There are v treatments each of which occurs in r blocks.

(iii) There can be established a relation of association between any two treatments satisfying the following requirements.

Given a set E of v elements, and given positive integers k, l (l ≤ k ≤ v), and λ, we understand by a tactical configurationC[k, l, λ, v] (briefly, configuration) a system of subsets of E, having k elements each, such that every subset of E having l elements is contained in exactly λ sets of the system.

A necessary condition for the existence of a configuration C[k, l, λ, v] is known (6) to be

This paper contains a number of results that grew out of an attempt to solve the following problem : Given a non-commutative ring R with suitable ascending chain condition, and a prime ideal P in R, to construct a corresponding local ring RP in which the extension P′ of P is a unique maximal prime, and to prove, if possible, that the intersection of the powers of P′ is zero. The present investigation is at best a preliminary attack on this problem since the contribution to the complete solution is comparatively small and the central problem of the intersection of the powers of P′ has not been touched.

The problem of determining the possible morphological types of convex polyhedra in three-dimensional Euclidean space E3 is well known to be quite hopeless. We lack not only any general way of determining whether there exists a convex polyhedron having as faces ƒ3 triangles, ƒ4 quadrangles, . . . , and ƒnn-gons, but even much more special questions of this kind seem to be rather elusive.

In a previous paper the author (1) has shown the following theorem.

Theorem A. If each of H and K is a real-valued bounded function of subintervals of the number interval [a, b] and m is a real-valued non-decreasing function on [a} b] such that each of the integrals

exists, then the integral

exists.

Let J be a Dedekind ring, F its quotient field, F′ a finite separable extension of F, and J′ the integral closure of J in F′. It has been shown by Artin (1) that a necessary and sufficient condition that J′ have an integral basis over J is that a certain ideal of F (namely, √(D/Δ), where D is the discriminant of the extension and Δ is the discriminant of an arbitrary basis of the extension) should be principal. More generally, he showed that if is an ideal of F′, then a necessary and sufficient condition that have a module basis over J is that N()√(D/Δ) should be principal.

Given an abelian group G and a mapping θ that maps a subgroup A of G homomorphically onto another subgroup B of G, then it is known (3) that there always exists an embedding group G* ⊇ G which is abelian and possesses an endomorphism θ* which coincides with θ on A, i.e. aθ = aθ* whenever aθ is defined. θ is called a partial endomorphism of G and θη a total endomorphism or simply an endomorphism that extends or continues θ.