In a previous investigation [1], the author has studied finite groups of an order g = pg0 where p is a prime and g0 an integer not divisible by p. This work has been continued by H. F. Tuan [5]. Let t denote the number of conjugate classes of which consist of element of order p. Tuan dealt with the groups for which t≦2 and which have a faithful representation of degree less than p - 1. We shall assume here that t≧3. We shall also suppose that does not have a normal subgroup of order p. We state here two results. We shall show (Corollary, Theorem 1) that if / is a faithful irreducible character of of degree n which has T>1 conjugates over the field of the g0-th roots of unity, then

We prove here an unpublished conjecture of Milnor which gives a complete set of multiplicative relations between the numbers

where ranges over complex roots of unity. Information of this type is useful in certain areas of topology as well as in number theory.

Let 5 denote the icosahedral group and let be the normalizer of a Sylow 5-subgroup of 5. Then the index of in 5 equals six. Let us represent 5 as a permutation group A on the set of residue classes of with respect to 5 Then it is clear that A is doubly transitive of degree 6 and order 60 = 2·5·6. Since 5 is simple, A does not contain a regular normal subgroup.

Let E and E′ be C∞ vector bundles over a C∞ manifold M. Denote by Γ(E) (resp. by Γ(E′) the vector space of C∞ cross-sections of E (resp. of E′) over M. Take a linear differential operator of the first order D: Γ(E) → Γ(E′) induced by a vector bundle mapping σ(D): jl(E) ′ E′, where Jk(E) denotes the vector bundle of k-jets of cross-sections of E.

We consider a group G of finite order g = pag′ where p is a prime number and (p, g′) = 1. Let Ω be the algebraic number field which contains the p-th roots of unity. Let K1, K2,…, Kn be the classes of conjugate elements in G and the first m(≦n) classes be p-regular. There exist n distinct (absolutely) irreducible characters x1, x2,…, xn of G.

In the arithmetic theory of automorphic functions on a symmetric bounded domain = G/K, as developed recently by Shimura and Kuga [2], [2a], it is important to consider a family of (polarized) abelian varieties on obtained from a symplectic representation ρ (defined over Q) of G (viewed as an algebraic group defined over Q) satisfying a certain analyticity condition. Recently, I have determined completely such representations, reducing the problem to the case where G is a Q-simple group and where ρ is a Q-primary representation ([3], [4]). It has turned out that, besides the four standard solutions investigated already by Shimura, there exist two more non-standard solutions, one of which comes from a spin representation of the orthogonal group and thus gives a family of abelian varieties on a domain of type (IV). The purpose of this short note is to explain how one can construct most simply, starting from the “regular representation” of the corresponding Clifford algebra, examples of such families, including also the non-analytic case.

The group of extensions Ext(A, Ga) and Ext(A, Gm) of an abelian variety A by the additive or multiplicative group Ga, Gm have been investigated in detail ([9], [10], [GACC]). On the other hand, F. Oort [8] and M. Miyanishi [6] recently studied the groups Ext(Ga, A) and Ext(Gm, A). The purpose of the present paper is to clarify the relationship between these groups. Our results show that the latter groups can be derived from the former.

It is well known that the semi-simple rings with minimum conditions coincide with the rings of global homological dimension zero and that the hereditary rings coincide with the rings of global dimension one. Eilenberg, Jans, Nagao and Nakayama gave some properties of hereditary rings in [4] and [11], which relate to global dimension of factor rings. As an example of non-commutative hereditary ring we know a tri-angular matrix ring over a semi-simple ring.

In 1952, the late Professor T. Nakayama succeeded in constructing the Galois theory for finite dimensional simple ring extensions [7]. And, we believe, the theory was essentially due to the following proposition: If a simple ring A is Galois and finite over a simple subring B then A is B′-A-completely reducible for any simple intermediate ring B′ of A/B [7, Lemmas 1.1 and 1.2]. Moreover, as was established in [5], Nakayama’s idea was still efficient in considering the infinite dimensional Galois theory of simple rings.

The many unanswerable questions (1) which arise in the study of finite groups have lead to a review of fundamental ideas, e.g. the Theorem of Burnside (3, p. 299; 2, 6) that if λ be any faithful irreducible representation of G over a field K, then every irreducible representation of G over K is contained in some tensor power of λ.

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.

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.

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.

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: (x1,…, xn) ∈ A × · · · × A → fλ(x1,…, 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(p1,…, pn) performed on other polynomials pi. An algebra A may satisfy a set R of identities p(x1,…, xr) = q(x1,…, xs), and then A shall be called an (F, R)-algebra.

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].

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).

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.

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)