This paper arose out of an effort to present a result which simplifies the application of the theory of blocks to what is often called the theory of exceptional characters. In order to obtain the most effective use of this result, it is necessary to reformulate part of this theory. Thus we present a development which explains an application of R. Brauer’s main theorem on generalized decomposition numbers. In particular, we improve a result of D. Gorenstein and the author [8; Proposition 25]. These results are needed in a forthcoming paper and will simplify somewhat the use of this theory in existing papers. We are interested principally in determining the values of certain irreducible characters on trivial intersection subsets. The organization of the theory presented here is influenced by an exposition of M. Suzuki [13] and uses concepts introduced by W. Feit and J. G. Thompson [7]. Also it is hoped that this exposition will serve as an introduction to the theory.

Resolutive compactification and harmonic measures. Let R be an open Riemann surface. A compact Hausdorff space R* containing R as its dense subspace is called a compactification of R and the compact set Δ = R* -R is called an ideal boundary of R. Hereafter we always assume that R does not belong to the class OG . Given a real-valued function f on Δ, we denote by the totality of lower bounded superharmonic (resp. upper bounded subharmonic) functions sonis satisfying

Ein projektiver Modul wird (in [1]) semi-perfekt genannt, wenn jedes epimorphe Bild von ihm eine projektive Hülle besitzt. Eine projektive Hülle eines Moduls C ist eine exakte Fole wobei P projektiv ist und der Kern Ke(f) von f Klein (= small = superflous)* in P ist. In [1] wird gezeigt, daB ein projektiver Modul P dann und nur dann semi-perfekt ist, wenn das Radikal Ra(P) von P klein in P ist, P̅ = P/Ra(P) halbeinfach ist und jede direkte Zerlegung von P̅ durch eine direkte Zerlegung von P induziert wird.

Einleitung: Die Ramanujan-Nagellshe Gleichung (Basis 2, Diskriminante—7).

Verallgemeinerung auf beliebige negative Diskriminanten D ≡ 1 mod. 23: Systematischer Ansatz zur Behandlung und erste notwendige Bedingungen für Lösbarkeit und Lösungen.

Endlichkeit der Lösungsanzahl.

Notwendige Bedingung für die Lösungen durch Kongruenzbetrachtung nach Potenzen von 2.

Notwendige Bedingung für die Lösungen durch Kongruenzbetrachtung nach Potenzen eines Primteilers von D.

Let G be a complex analytic group, and let A be the representation space of a finite-dimensional complex analytic representation of G. We consider the cohomology for G in A, such as would be obtained in the usual way from the complex of holomorphic cochains for G in A. Actually, we shall use a more conceptual categorical definition, which is equivalent to the explicit one by cochains. In the context of finite-dimensional representation theory, nothing substantial is lost by assuming that G is a linear group. Under this assumption, it is the main purpose of this paper to relate the holomorphic cohomology of G to Lie algebra cohomology, and to the rational cohomology, in the sense of [1], of algebraic hulls of G. This is accomplished by using the known structure theory for complex analytic linear groups in combination with certain easily established results concerning the cohomology of semidirect products. The main results are Theorem 4.1 (whose hypothesis is always satisfied by a complex analytic linear group) and Theorems 5.1 and 5.2. These last two theorems show that the usual abundantly used connections between complex analytic representations of complex analytic groups and rational representations of algebraic groups extend fully to the superstructure of cohomology.

The basic properties of associative rings R satisfying a polynomial identity p[x 1…, xn ] = 0 were obtained under the assumptions that the ring was an algebra [e.g., [4] Ch. X], or with rather strong restrictions on the ring of operators ([1]). But it is desirable to have these properties for arbitrary rings, and the present paper is the first of an attempt in this direction. The problem is almost trivial for prime or semi-prime rings but quite difficult in arbitrary rings. The known proofs for algebras have to be modified and in some cases new proofs have to be obtained as the existing proofs fail to exploit the known structure. In the present paper we extend the results of [1] on the nil subalgebras of a ring with an identity for arbitrary multiplicative nil semi-groups of the ring and for arbitrary rings.

Let G be a finite group of odd order with an automorphism θ of order 2. (We use without further reference the fact, established by W. Feit and J. G. Thompson, that all groups of odd order are soluble.) Let Gθ denote the subgroup of G formed by the elements fixed under θ. It is an elementary result that if Gθ = 1 then G is abelian. But if we merely postulate that G θ be cyclic, the structure of G may be considerably more complicated—indeed G may have arbitrarily large soluble length.

We prove in this paper a theorem on analytic mappings of the complex space Cn into the complex projective space Pn . The theorem is closely related to that of S. S. Chern in [1], and the main idea of the proof is the same with the latter, though the calculations are rather different. The background of our calculation is the normal contact metric structure which was found by S. Sasaki and Y. Hatakeyama [4].

In the present paper by an algebraic system (algebra) A we shall mean a system with a set F of operations fλ : (x 1,…, xn ) ∈ A × · · · × A → fλ (x 1,…, xn ) ∈ A. A polynomial p(x1, …, xr ) is a function of variables x1 ,…, xr which is either one of the xi , or (recursively) a result of some operation fλ (p 1,…, pn ) performed on other polynomials pi . An algebra A may satisfy a set R of identities p(x1 ,…, xr ) = q(x 1,…, xs ), and then A shall be called an (F, R)-algebra.

Let k be an algebraic number field of finite degree, K be its normal extension of degree n, and ŝ be the set of those primes of K which have degree 1. Using this set s instead of the set of all primes of K, we define an s-restricted idèle of K by the same way as ordinary idèles. It is known by Bauer that the normal extension of an algebraic number field is determined by the set of all primes of the ground field which are decomposed completely in the extension field. This suggests that if we treat abelian extensions over K which are normal over k, the class field theory is expressed by means of the ŝ-restricted idèles (theorem 2). When K = k, ŝ is the set of all primes of K, and we have the ordinary class field theory.

By focussing attention on indecomposable modular representations J. G. Thompson [11] has recently simplified and generalized some classical results of R. Brauer [1] concerning groups which have a Sylow group of prime order. In this paper this approach will be used to prove some results which generalize theorems of R. Brauer [2] and H. F. Tuan [12].

1. Concerning a positive lower semicontinuous kernel G on a locally compact Hausdorff space X the following existence theorem was obtained in [3].

Let R be an integral domain, Q its quotient field. For any R-module A we write d(A) for its homological dimension (minimal length of a projective resolution), or dR(A) if it is necessary to call attention to R.

The hyperoctahedral group Gn of order 2nn! is generated by permutations and sign changes applied to n digits, d = 1, 2,…, n. The 2n sign changes generate a normal subgroup ∑n whose factor group Gn/∑n is isomorphic with the symmetric group Sn of order n!. To each irreducible orthogonal representation ‹X; μ› of Gn corresponds an ordered pair of partitions [λ] of l and [μ] of m, where l+m = n.

In this paper we shall give a definition of an analogue for Jordan algebras of the classical notion of a Cartan subalgebra of a Lie algebra. This is based on a notion of associator nilpotency of a Jordan algebra. A Jordan algebra is called associator nilpotent if there exists a positive (odd) integer M such that every associator of order M formed of elements of is 0 (§2).

It is known classically that abelian varieties of dimension one over the field of complex numbers may be expressed by non-singular Hesse’s canonical cubic plane curves, The purpose of the present paper is to generalize this idea to higher dimensional case.

In a previous paper [4], we studied a class of algebras over a commutative ring R which we called semisimple algebras. Here we shall study simple algebras.

1. When a permutation group G on a set Ω is given, a transitive extension G of G is defined to be a transitive permutation group on the set Γ which is a union of Ω and a new point ∞ such that the stabilizer of ∞ in G1 is isomorphic to G as a permutation group on Ω. The purpose of this paper is to prove that many known simple groups which can be represented as doubly transitive groups admit no transitive extension. Precise statement is found in Theorem 2. For example, the simple groups discovered by Ree [5] do not admit transitive extensions. Theorem 2 includes also a recent result of D. R. Hughes [3] which states that the unitary group U3 (q) q>2 does not admit a transitive extension. As an application we prove a recent theorem of H. Nagao [4], which generalizes a theorem of Wielandt on the non-existence of 8-transitive permutation groups not containing the alternating groups under Schreier’s conjecture.

1. Let Ω be a finite set of arbitrary elements and let (G, Ω) be a permutation group on Ω. (This is also simply denoted by G). Two permutation groups (G, Ω) and (G, Γ) are called isomorphic if there exist an isomorphism σ of G onto H and a one to one mapping τ of Ω onto Γ such that (g(i)) τ=gσ(iτ) for g ∊ G and i∊Ω. For a subset Δ of Ω, those elements of G which leave each point of Δ individually fixed form a subgroup G Δ of G which is called a stabilizer of Δ. A subset Γ of Ω is called an orbit of G Δ if Γ is a minimal set on which each element of G induces a permutation. A permutation group (G, Ω) is called a group of rank n if G is transitive on Ω and the number of orbits of a stabilizer Ga of a ∊ Ω, is n. A group of rank 2 is nothing but a doubly transitive group and there exist a few results on structure of groups of rank 3 (cf. H. Wielandt [6], D. G. Higman M).

Let A well known theorem of Hardy and Ramanujan states: v(n) = (1 = 0(1))log log n holds for all n if we neglect a sequence of density 0 [5]. Define for 2 ≤j≤v(n)