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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 R1. . ., 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].

In order to extend Nevanlinna’s first and second fundamental theorems to arbitrary analytic mappings between Riemann surfaces, Sario [8, 9] introduced a kernel function on an arbitrary Riemann surface generalizing the elliptic kernel on the Riemann sphere. Because of the importance of the potential theoretic method in the value distribution theory, we discussed potentials of Sario’s kernel in [4]. In that paper the validty of Frostman’s maximum principle for Sario’s potentials was left unsettled. The main object of this paper is to resolve this question (Theorem 1). As a consequence the fundamental theorem of the potential theory is obtained in its complete form for Sario’s potentials (Theorem 2).

Let R be a field and S a separable algebraic closure of R with galois group R. In [8] Harrison succeeded in describing R/′R in terms of R only. More precisely, he constructed a certain complex (R, Q/Z) and proved Homc, where Homc denotes continuous homomorphisms and H2 stands for the second cohomology group of the complex . In this paper, which is mainly expository in nature, we reexamine Harrison’s proof and show how [8] connects with Kummer theory and the theory of galois algebras [16]. We emphasize that most of the ideas on which this paper is based originate in [8].

The classification of Riemann surfaces has largely reached its completion. The purpose of the present paper is to lay the foundation for a new intriguing field in the classification theory: Riemannian spaces with Euclidean metrics. The paper is self-contained, both for the Riemann surface expert and the reader whose main interest is with higher dimensions. The significance of locally Euclidean spaces lies, first of all, in that their function-theoretic nature differs for dimensions n>2 and n=2. The existence or nonexistence of Green's functions and positive or bounded harmonic functions in Rn, punctured Rn, and in the punctured flat torus offer simple examples. A striking phenomenon is that, despite such differences, the basic inclusion relations remain valid. Moreover, capacities and null-classes can be defined for components of point sets in Rn.

Main purpose of the present paper is to study formal deductions described along the line of my former work [1]. In the present paper, I restrict myself to the primitive logic. To extend this method to other logics such as the lower classical predicate logic or the intuitionistic predicaste logic, my way of practical discription has to undergo a certain extent of modification.

The present paper is devoted to the study of differential geometry of Kaehlerian homogeneous spaces. In section 1 we deal with the canonical decomposition of a simply connected complete Kaehlerian space and that of its largest connected group of automorphisms. We know that a simply connected complete Riemannian space V is the product of Riemannian spaces V0, V1, …, Vn, where V0 is a Euclidean space and V1, …,Vn are not locally flat and their homogeneous holonomy groups are irreducible [2]. Moreover, if V is homogeneous, so are all Vk [10]. We shall show that if V is Kaehlerian space with real analytic metric (resp. Kaehlerian homogeneous space), each factor Vk is also Kaehlerian (resp. Kaehlerian homogeneous) and that V is the product of V0, V1, …, Vn as Kaehlerian space. We call this decomposition the de Rham decomposition of the Kaehlerian space V. Although this result is supposedly known, there is no published proof as yet. Using this decomposition theorem we shall show that the largest connected group of automorphisms of a simply connected complete Kaehlerian space with real analytic metric is the direct product of those of the factors of the de Rham decomposition. In the Riemannian case this result has be been established in [3] by one of the authors of the present paper.

In the previous papers [1], [2] we have studied the prolongations of G-structures to tangent bundles of arbitrary order, and in [3] we have considered the prolongations of connections to the tangential fibre bundles of higher order. The purpose of the present paper is to study the liftings of tensor fields and affine connections to tangent bundles of higher order. In fact, most of results in [4], [5] will be generalized in a natural fashion and some of the formulas concerning vertical and complete lifts in [5] will be unified and generalized in our formulas concerning ‹Λ›-lifts (cf. §3).

It is the purpose of the present paper to give a natural method of realizing bounded symmetric domains as matrix-spaces. Our method yields, as special cases, the well-known bounded models of irreducible bounded symmetric domains of classical type (I)-(IV), as were already described in the original paper of E. Cartan [1] (see §3; we follow in this paper the classification table in [14], not in [1]). A direct application of this method will be to determine the canonical bounded models of the irreducible bounded symmetric domains of exceptional type; it will be published in another paper (see [6], [7] for the summary of the results).

The theory of stochastic differential equations in a differentiate manifold has been established by many authors from different view-points, especially by R Lévy [2], F. Perrin [1], A. Kolmogoroff [1] [2] and K. Yosida [1] [2]. It is the purpose of the present paper to discuss it by making use of stochastic integrals.

In [5], Ikeda-Nagao-Nakayama gave a characterization of algebras of cohomological dimension ≦n In a subsequent paper [4] Eilenberg gave an alternative treatment of the same question. The present paper is devoted to the discussion of a number of questions suggested by the results of [4] and [5]. Among others it is shown that the conditions employed in stating the main results in [4] and [5] are equivalent, so that the main results of these two papers are in accord. Further, the cohomological dimension of a residue-algebra is studied in terms of that of the original algebra and the (module-) dimension of the associated ideal. The terminology and notation employed here are that of [3].

Let S denote the integral closure of a complete discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, G the Galois group of the quotient field extension, and f an element of Z2(G,U(S)) where U(S) denotes the multiplicative group of units of S. A crossed product Δ(f, S, G) whose radical is generated as a left ideal by the prime element II of S is an hereditary order according to the Corollary to Thm. 2. 2 of [2], and we call such a crossed product a II-principal hereditary order. In previous papers the author has studied II-principal hereditary orders Δ(f, S, G) for tamely and wildly ramified extensions S of R (see [10] and [11]). The purpose of this paper is to study II-principal hereditary orders Δ(f, S, G) with no restriction on the extension S of R.

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.

I. Introduction. The purpose of this paper is to discuss functions defined on the continuous sample paths of Gaussian Markov processes which serve as Green’s functions for pairs of generalized Schroedinger equations. The results extend the author’s earlier paper [2] to a forward time version, and consider different boundary conditions.

We consider in this paper eleven conditions on a commutative ringR. The first of these is thatRcontains an identity. It is well known that each of the other properties is a consequence of the first condition. This paper considers other relations which exist between these properties. A complete diagram of all simple implications which exist between the eleven properties, together with proof of these implications, is given in section 3. Examples illustrating simple implications which do not hold are presented in section 4.

Let A = R/, where R is a regular local ring of arbitrary dimension and is an ideal of R. If A is a Gorenstein ring and if height = 2, it is easily proved that A is a complete intersection, i.e., is generated by two elements (Serre [5], Proposition 3). Hence Gorenstein rings which are not complete intersections are of embedding codimension at least three. An example of these rings is found in Bass’ paper [1] (p. 29). This is obtained as a quotient of a three dimensional regular local ring by an ideal which is generated by five elements, i.e., generated by a regular sequence plus two more elements. In this paper, suggested by this example, we prove that if A is a Gorenstein ring and if height = 3, then is minimally generated by an odd number of elements. If A has a greater codimension, presumably there is no such restriction on the minimal number of generators for , as will be conceived from the proof.

Throughout this paper all rings are assumed to have unit elements. A ring Λ is said to be semi-primary if its Jacobson radical N is nilpotent and Г = Λ/N satisfies the minimum condition. The main objective of this paper isTHEOREM I. Let A be a semi-primary algebra over a field K. Let N be the radical of Λ and Г = Λ/N. IfThendim Λ = gl.dim Λ.

Recently the writer has obtained some results concerning meromorphic or algebroidal functions with the set of essential singularities of capacity zero, with an aid of a theorem of Evans. In the present paper, suggested from recent interesting papers of Sario and Pfluger, the writer will extend his results to single-valued analytic functions defined on open abstract Riemann surfaces with null boundary in the sense of Nevanlinna, using a lemma instead of Evans’ theorem.

As a generalization of the notion of Frobenius algebras over a field Kasch [103 introduced that of Frobenius extensions of a ring. The present writers [13] recently freed one of Kasch’s main theorems from its rather strong S-ring assumption of the ground ring. However, even with the removal of the S-ring assumption of the ground ring the notion does not seem general enough, and we wish, in the present paper and its sequel, to develope the theory upon the basis of a more general notion of Frobenius extensions. Thus, we replace the free module property of the extension by the projective module property (according to a general tendency in algebra), which has been done in fact in case of Frobenius algebras over a commutative ring in a previous work by Eilenberg and one of the writers [4], and, further, take automorphisms of the ground ring into the definition of Frobenius extensions (which seems quite natural particularly in case of non-commutative rings). To such generalized notion of Frobenius extensions we may extend many of Kasch’s theorems, including those which are immediate extensions of classical theorems for Frobenius algebras and those which are essentially new, as the above alluded endomorphism ring theorem. Also homological properties of Frobenius extensions, as were developed in Hirata’s [6] recent paper in succession to Eilenberg-Nakayama [4], can be extended to our present generalized case; we shall also exceed [4], [6] somewhat in considering injective and weak dimensions.

In [0], the writer proved some theorems of Hasse type for two orthogonal groups which operate on the same vector space. In this paper, we shall further generalize those results in two directions. One is to consider the propositions of that type for two orthogonal groups which operate respectively on two vector spaces whose dimensions are different from each other, and the other is to deal with some conspicuous subgroups of an orthogonal group simultaneously which play important roles in the structure theory for orthogonal groups. For this reason, the present paper consists of three steps §1, §2 and §3 which give the generalizations in the above sense of the results in the corresponding sections of [0].

This paper contains part of the author’s Ph.D. dissertation directed by Frederick Bagemihl at Wayne State University. The research was supported by a grant from the Michigan Institute of Science and Technology.Alice Roth has made an extensive study of entire meromorphic functions with prescribed behavior along half rays emanating from the origin (6). The question arose whether analogous results could be found for functions meromorphic in the unit disk with the same behavior prescribed along an exhaustive class of spirals emanating from the origin. In this paper, I present a class of spirals which satisfactorily fills this role. However, I make no claim to the effect that only this class will suffice.

In the previous paper [6], we studied the liftings of tensor fields to tangent bundles of higher order. The purpose of the present paper is to generalize the results of [6] to the tangent bundles of pr-velocities in a manifold M— notions due to C. Ehresmann [1] (see also [2]). In §1, we explain the pr-velocities in a manifold and define the (Λ)-lifting of differentiable functions for any multi-index λ -(λ1, λ2,…,λp) of non-negative integers λi satisfying ΣΛi≤r. In § 2, we construct ‹Λ›-lifts of any vector fields and ‹Λ›-lifts of 1-forms. The ‹Λ›-lift is a little bit different from the ‹Λ›-lift of vector fields in [6].

The concept of semi-local rings was introduced by C. Chevalley [1], which the writer has generalized in a recent paper [7] by removing the chain condition. The present paper aims mainly at the study of completions of semi-local rings. First in § 1 we investigate semi-local rings which are subdirect sums of semi-local rings, and we see in § 2 that a Noetherian semi-local ringRis complete if (and only if)R/pis complete for every minimal prime divisorpof zero ideal, together with some other properties. Further we consider in § 3 subrings of the completion of a semi-local ring. § 4 gives some supplementary remarks to [7], Chapter II, Proposition 8.

1. Introduction. In this paper we develop a method to find all the irreducible and inequivalent representations over the complex field C of a family of finite groups that includes the metabelian groups. The outline of the paper is as follows: In §2 we let P be a one-dimensional representation of a subgroup H of a finite group G and find a maximal subgroup K, H ⊆ K ⊆ G, such that an extension P̅ of P to K exists. We show that the induced representation P̅G is irreducible if K is normal in G. In §3 we give all the irreducible and inequivalent representations of the “generalized metabelian group” G, and in particular of the metabelian group, and decompose the group ring CG into its simple components. The convenience of this method is shown in §4 where we determine the representations of the metacyclic group and a metabelian group of order 2sp2, p an odd prime and s|p — l. The algorithm in §5 is supplementary and can be used to find representations of more general groups than the two in §4.

We have several definitions of the positivity of a vector bundle, differentiate definitions, an algebro-geometric definition, a topological definition etc. In § 1 we review the definitions and the relations between them. For a line bundle all the definitions are equivalent and every one agrees that they are reasonable. For a vector bundle, however, the definitions are not necessarily equivalent. One of the main results of this paper is the equivalence of the definitions over a complete non-singular curve. The proof is given in §2. We proved this over an elliptic curve in Umemura [18]. In this case the proof was based on Atiyah’s classification. To prove the equivalence over a curve of genus ≥ 2, the fundamental lemma is; A stable bundle of positive degree is positive in the sense of Nakano. The tool used to prove this lemma is the theory of stable bundles due to Narasimhan and Seshadri [11] —they establish a correspondence between stable bundles and certain types of irreducible unitary representations of a Fuchsian group.

Let X = (X(t); t ∈ R) be a real stationary mean continuous Gaussian process with expectation zero which is purely nondeterministic. In this paper we shall investigate the structure of splitting fields of X having finite multiple Markovian property using the results in [6]. We follow the notations and terminologies in [6].

The theory of classification of homogeneous bounded domains in the complex number space Cn has been developed mainly in the recent papers [10], [6], [3] and [7]. As a result, the classification is reduced to that of S-algebras due to Takeuchi [7] which correspond to irreducible Siegel domains of type I or type II (For the definition of irreducibility see § 1). On the other hand Pjateckii-Sapiro [5] found large classes of homogeneous Siegel domains obtained from classical self-dual cones. Even in lower-dimensional cases, however, there are still homogeneous Siegel domains which do not appear in his results.

In [7], Matsushima studied the vector bundles over a complex torus. One of his main theorems is: A vector bundle over a complex torus has a connection if and only if it is homogeneous (Theorem (2.3)). The aim of this paper is to prove the characteristic p > 0 version of this theorem. However in the characteristic p > 0 case, for any vector bundle E over a scheme defined over a field k with char, k = p, the pull back F*E of E by the Frobenius endomorphism F has a connection. Hence we have to replace the connection by the stratification (cf. (2.1.1)). Our theorem states: Let A be an abelian variety whose p-rank is equal to the dimension of A. Then a vector bundle over A has a stratification if and only if it is homogeneous (Theorem (2.5)).

We treat the sum , where α and γ are real with α positive. This sum was treated first by Hardy and Littlewood [4], and after them, by Behnke [1] and [2], Mordell [9], Wilton [11] and others. The reader will find its history in [7] and in the comments of the Collected Papers [4]. Here we show that the sum can be expressed explicitly, together with an error term O(N1/2), using the regular continued fraction expansion of α. As the statements have complications we will divide them into two theorems. In the followings all letters except ϑ, i, σ, ζ, χ and those in 3° are real, N is a positive real, and always k, n, a, A, B, C, D and E denote integers. The author expresses his thanks to Professor Tikao Tatuzawa and Professor Tomio Kubota for their encouragements.

Let k be a p-adic number field and o be the ring of all integers of k. Let K/k be a cyclic ramified extension of prime degree p with Galois group G. Then the ring of all integers of K is o[G]-module. The purpose of this paper is to give a necessary and sufficient condition for o[G]-modale to be indecomposable.

Consider a tower of principal fibrationsThat is, Ei+1 is the pullback of Ei→Ri and the path fibration PRi → Ri. The question arises as to whether or not the tower can be shortened, that is, whether or not En+1→ B is fiber homotopically equivalent to a nice fibration E → B. If “nice” is taken to mean “principal” then sufficient conditions are known. They involve connectivity assumptions on the Ei. In this paper “nice” is taken to mean “D-relatively principal” for some space D.

It is well-known that an Anosov diffeomorphism f on a compact manifold is structurally stable in the space of all C1-diffeomorphisms, with the C1-topology (Anosov [1]). In this paper we show that f is also structurally stable in the space of all lipeomorphisms, with a lipschitz topology. The proof is similar to that of the C1-case by J. Moser [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.

In the papers [11] and [18] Rohlin and Thom have introduced an equivalence relation into the set of compact orientable (not necessarily connected) differentiable manifolds, which, roughly speaking, is described in the following manner: two differentiable manifolds are equivalent (cobordantes), when they together form the boundary of a bounded differentiable manifold. The equivalence classes can be added and multiplied in a natural way and form a graded algebra Ω relative to the addition, the multiplication and the dimension of manifolds. The precise structures of the groups of cobordism Ωk of dimension k are not known thoroughly. Thom [18] has determined the free part of Ω and also calculated explicitly Ωk for 0 ≦ k ≦ 7.

Let k denote the quotient field of a complete discrete rank one valuation ring R. The purpose of this paper is to establish a relationship between the Brauer group of k and the set of maximal orders over R which are equivalent to crossed products over tamely ramified extensions of R.

It is our object in this paper to show that the recent results of K. Ito and M. Nisio [4] on the oscillation function of Gaussian processes on [0,1] are valid for Gaussian processes with a general multiparameter “time” set T. Except in extending Theorem 4 of [4] where we assume T to be the d-dimensional cube, in all other cases we allow T to be a separable metric space. Despite the generality of the time set, the proofs are achieved essentially using the method of the above mentioned authors. However, in Theorem 1 below we find the use of Lemma 6 of [5] more convenient than the approach via orthogonal expansions and Kolmogorov’s zero-one law as is done in [4].

The purpose of this paper is to give the limit formula of the Kronecker’s type for a non-holomorphic Eisenstein series with respect to a Hubert modular group in the case of an arbitrary algebraic number field. Actually, we shall generalize the following result which is well-known as the first Kronecker’s limit formula. From our view-point, this classical case is corresponding to the case of the rational number field Q.

If GK is a Chevalley group over a field K of prime characteristic p, the irreducible representations of GK over K form a natural object of study. The basic results have been obtained by Steinberg [15], who showed that, if K is perfect, then each irreducible rational representation of GK over K is a tensor product of representations obtained from certain basic representations by composing them with field automorphisms. These basic representations were obtained by “integrating” the irreducible restricted representations of a restricted Lie algebra associated with the group, which had been studied earlier by Curtis [7]. The present author had obtained the main results previously for the groups SL(n, K), Sp(2n, K) by different means, involving reduction (mod p) from the characteristic 0 case [16]. In this paper we extend this method to the other types of groups, in the hope that some additional insight may be gained.

Beurling and Deny [1], [2] introduced Dirichlet spaces by generalizing the notion of“energy”in potential theory. They showed the existences of balayaged measures and condensor measures in the theory of Dirichlet spaces. The purpose of this paper is to characterize the supports of those measures. First we obtain the following result.

Let S be the integral closure of a discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, and denote the Galois group of the quotient field extension by G. It has been proved by Auslander and Rim in [4] that the trivial crossed product Δ(l, S, G) is an hereditary order for tamely ramified extensions S of R and that Δ(l, S, G) is a maximal order if and only if S is an unramified extension of R. The purpose of this paper is to study the crossed product Δ(f, S, G) where [f] is any element of H2(G, U(S)) and S is a tamely ramified extension of R with multiplicative group of units U(S).

In a recent paper [2], an operator valued function space integral was defined by the authors as follows.

Throughout this paper, a space means a T1-space. A space is called fully normal if every open covering of it has a Δ-refinement , that is, an open covering for which the stars (x, ) form a covering which refines . A space is called paracompact if every open covering of it has a locally finite (= neighborhood finite) open covering which refines . It is well known that paracompactness is identical with full normality in a Hausdorff space ([3], [7]).

Some important notions in the theory of binary relations such as the relative product of two relations and the converse of a relation are defined in Whitehead and Russell’s “Principia mathematica” ([1]). McKinsey ([2]) and Tarski ([3]) gave their systems of postulates for the calculus of relations. Ore studied on equivalence relations ([4]) and Riguet on closures (fermeture) of relations ([5] and [6]), and they obtained remarkable results on the structure of these relations. The purpose of this paper is to examine the relative properties of some relations to each other, whose notions are closely related to the notions given by Riguet.

Let G be a connected semi-simple algebraic group defined over Q and let Γ be a discrete subgroup of GR (the subgroup of G consisting of points rational over R) such that Γ\GR is compact. The main purpose of the present paper is to prove that for a certain type of group G there exists an invariant algebraic differential from ω on G of highest degree defined over Q such that

The purpose of this paper is to get a certain explicit expression of automorphic factors, formulated rather differently than usual, of the classically well known theta function(1)

Let k(≧3) be a positive integer and φ(k) be the Euler function. We denote by % one of the φ(k) characters formed with modulus k, and by % the principal character. Let Lis, %) be the L-series corresponding to % Throughout the paper we use c and c(ε) to denote respectively an absolute positive constant and a positive constant depending on parameter ε( >0) alone, not necessarily the same at their various occurrences. We use the symbol y = O(Z) for positive X when there exists a c satisfying Y ≦ in the full domain of existence of X and Y.

Throughout this paper F denotes a field complete with respect to a discrete valuation, kF the residue field of F, K/F a finite Galois extension with Galois group G = G(K/F). The ring of integers 0K of K contains the (unique) prime ideal ; the collection of ideals n for all integers n are ambiguous ideals i.e. G-modules.

In this paper, we shall show that any Gaussian measure on a separable or reflexive Banach space is an abstract Wiener measure in the sense of L. Gross [1] and, for the proof of that, establish the Radon extensibility of a Gaussian measure on such a Banach space. In addition, we shall give some remarks on the support of an abstract Wiener measure.

Throughout this paper all functions are single-valued. Let R be a Riemann surface. We shall denote by φ∧ the least harmonic majorant of a function φ defined in R if it has the meaning.

The purpose of this paper is to investigate a quasi-flow which is a one-parameter group of non-singular measurable point transformations on a measure space. If, in particular, the transformations are all measure preserving (i.e. a flow is given), the ergodicity together with the mixing property, the spectral or metrical type, increasing partitions of the space and the entropy of the flow are our main interests. Those methods used in the study of a flow are frequently useful for our approach. For example, the concept of a special flow introduced by W. Ambrose [3] plays an important role and the representation of a given flow in terms of a special flow is a powerful tool in the study of flows. L.M. Abramov [1] calculates the entropy of a flow with the help of the representation. As another example we give attention to the work of G. Maruyama [10] and H. Totoki [15] where they discuss a general time-change of flows the basic idea of which was originated by E. Hopf [8]. They discuss the invariant measure of a general time-changed flow and prove that the ergodicity is inherited and the entropy is kept invariant by the time-change. In the study of quasi-flows, we shall use both the repesentation in terms of a special quasi-flow and a time-change. Besides a quasi-flow requires its own methods in the investigation and it gives us some further problems such as the existence of an invariant measure (cf. [4], [5], [6] and [7]) and related topics.

In [1] R. Brauer asked the following question: Letbe a finite group,pa rational prime number, andBa p-block ofwith defectdand defect group. Is it true thatis abelian if and only if every irreducible character inBhas height 0 ? The present results on this problem are quite incomplete. Ifd-0, 1, 2 the conjecture was proved by Brauer and Feit, [4] Theorem 2. They also showed that ifis cyclic, then no characters of positive height appear inB. Ifis 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 divisorpof the order of. Actually, one half of this has already been proved in [7]. There it was shown that ifis a p-solvable group, wherepis any prime, and ifis abelian, then the condition on the irreducible characters inBis satisfied.