In [1] R. Brauer asked the following question: Let be a finite group, p a rational prime number, and B a p-block of with defect d and defect group . Is it true that is abelian if and only if every irreducible character in B has height 0 ? The present results on this problem are quite incomplete. If d-0, 1, 2 the conjecture was proved by Brauer and Feit, [4] Theorem 2. They also showed that if is cyclic, then no characters of positive height appear in B. If is normal in , the conjecture was proved by W. Reynolds and M. Suzuki, [12]. In this paper we shall show that for a solvable group , the conjecture is true for the largest prime divisor p of the order of . Actually, one half of this has already been proved in [7]. There it was shown that if is a p-solvable group, where p is any prime, and if is abelian, then the condition on the irreducible characters in B is satisfied.

Let H be a normal subgroup of a finite group G, and let ζ be an (absolutely) irreducible character of H. In [7], Clifford studied the irreducible characters X of G whose restrictions to H contain ζ as a constituent. First he reduced this question to the same question in the so-called inertial subgroup S of ζ in G, and secondly he described the situation in S in terms of certain projective characters of S/H. In section 8 of [10], Mackey generalized these results to the situation where all the characters concerned are projective.

Our purpose here is to study the irreducible representations of semisimple algebraic groups of characteristic p 0, in particular the rational representations, and to determine all of the representations of corresponding finite simple groups. (Each algebraic group is assumed to be defined over a universal field which is algebraically closed and of infinite degree of transcendence over the prime field, and all of its representations are assumed to take place on vector spaces over this field.)

Let K be a field of characteristic zero. Let V be an n-dimensional vector space over K and let S be the graded ring of polynomial functions on V. If G is a group of linear transformations of V, then G acts naturally as a group of automorphisms of S if we define

The elements of S invariant under all γ ∈ G constitute a homogeneous subring I(S) of S called the ring of polynomial invariants of G.

In the previous paper [3] the author has shown a possibility to construct a series of sfields by taking sfields of quotients of split crossed products of a sfield. In this paper the same problem is treated, and, by considering general crossed products, a generalization of the previous result is given: Let K be a sfield and G be the join of a well-ordered ascending chain of groups Gα of outer automorphisms of K such that a) G1 is the identity automorphism group, b) Gα is a group extension of Gα-1 by a torsion-free abelian group for each non-limit ordinal α, and c) for each limit ordinal α. Then an arbitrary crossed product of K with G is an integral domain with a sfield of quotients Q and the commutor ring of K in Q coincides with the centre of K.

In [3] R. Brauer gave a proof of his theorem on generalized decomposition numbers which was first announced in [1], and a simplification of it has been made by K. Iizuka [5]. In this note we shall show that the theorem may be proved from another point of view by using some results obtained by J. A. Green in [4].

After stating some results by Green and Osima in the first and second sections we first prove a theorem on characters (Theorem 1) and by using the theorem we prove Brauer’s theorem in the fourth section.

Let be a permutation group on n-letters 1, 2,… ,n . Let be the subgroup of fixing suitable one letter, say 1. For any element G of , a non-singular matrix G* = (gij) of degree n is defined by the equation

The natural numbers play a part in the formulation of logical syntax inasmuch as they are used to count the symbols in a sentence, or the sentences in a proof, etc. In the present paper, we shall study an infinitary logical calculus which is based on replacing the ordinary natural numbers in the capacity just mentioned, by a non-standard model of arithmetic. (Compare refs. 3, 5, 6 for some other logical calculi of infinitary nature). Thus, our calculus will include formulae of length n for any natural number n, finite or infinite, in the chosen non-standard model of arithmetic. Evidently, the study of such formulae can be of interest only if we introduce concepts which are beyond the power of expression of the notions borrowed from the standard case. It turns out that the concept of truth in a model, when defined by means of Skolem functions has this character and involves a curious phenomenon which is analogous to one first pointed out by H. Steinhaus and J. Mycielski for another kind of infinitary language. Thus, while in the standard predicate calculus the negation of a sentence in prenex normal form is reduced to prenex normal form by changing the type of the quantifiers and by shifting the sign of negation, this procedure is not legitimate when truth is defined in this way in our calculus.

We have introduced in our former work [1] a theory of mathematical objects which can be regarded as a prototype of set theory. We have been successful to imbed the Zermelo set-theory [3] without the axiom of choice in our system. However, it seems impossible to imbed the Fraenkel set-theory [4] in our system even without the axiom of choice. In this work, we introduce another system of object theory in which we can imbed the Fraenkel set-theory without the axiom of choice. We shall denote our former system by OZ (object theory in the manner of the Zermelo set-theory) and the new system we are going to introduce in this work by OF (object theory in the manner of the Fraenkel set-theory). We shall also denote the Zermelo set-theory without the axiom of choice by SZ and the Fraenkel set-theory without the axiom of choice by SF.

One of the most powerful tools in homotopy theory is the homotopy groups of a triad introduced by Blakers and Massey in [1]. Our aim here is to develop systematically the formal, elementary aspects of the theory of a generalized triad and the mapping track associated with it. This will be used in §5 to deduce a result (Theorem 5.5) which seems to be closely related to an exact sequence established by Brown [2].

Let E be a totally-disconnected compact set in the z-plane and let Ω be its complement with respect to the extended z-plane. Then Ω is a domain and we can consider a single-valued meromorphic function w = f(z) on Ω which has a transcendental singularity at each point of E. Suppose that E is a null-set of the class W in the sense of Kametani [4] (the class NB in the sense of Ahlfors and Beurling [1]). Then the cluster set of f(z) at each transcendental singularity is the whole w-plane, and hence f(z) has an essential singularity at each point of E. We shall say that a value w is exceptional for f(z) at an essential singularity ζ ∈ E if there exists a neighborhood of ζ where the function f(z) does not take this value w.

Let w = w(z) be any K-quasiconformai mapping (in the sense of Pfluger-Ahlfors) of a domain D in the z-plane into the w-plane. Since w = w(z) is a measurable mapping (vid. Bers [1]), it transforms any set of Hausdorffs 2-dimensional measure zero in D into such another one. However, A. Mori [5] showed that for 0 < α ≤ 2, any set of -dimensional measure zero in a Jordan domain D is transformed by w = w(z)into a set of α-dimensional measure zero.

Some years ago, Kuramochi gave in his paper [5] a very interesting theorem, which can be stated as follows.

THEOREM OF KURAMOCHI. Let R be a hyperbolic Riemann surface of the class Of OHR(OHD,resp.). Then, for any compact subset K of R such that R—K is connected, R—K as an open Riemann surface belongs to the class 0AB(OAD resp.).

In this paper R is a commutative noetherian local ring with unit element 1 and M is its maximal ideal. Let K be the residue field R/M and let {t1,t2,…, tn) be a minimal system of generators for M. By a complex R<T1. . ., Tp> we mean an R-algebra* obtained by the adjunction of the variables T1. . ., Tp of degree 1 which kill t1,…, tp. The main purpose of this paper is, among other things, to construct an R-algebra resolution of the field K, so that we can investigate the relationship between the homology algebra H (R < T1,…, Tn>) and the homological invariants of R such as the algebra TorR(K, K) and the Betti numbers Bp = dimk TorR(K, K) of the local ring R. The relationship was initially studied by Serre [5].