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The theory of modular representations of the symmetric group was studied first by Nakayama (5, 6), and later by Thrall and Nesbitt (11) and Robinson (7, 8, 9). Nakayama built up his elaborate theory of hooks for the express purpose of studying this problem, while Robinson's extensive work on the various phases of the relationship between Young diagrams, skew diagrams and star diagrams on the one hand, and representations of the symmetric group on the other, culminating in a set of relations among the degrees of the representations, serves as a starting point for this paper.

In the present paper, the authors extend the Cousin theorems and the continuity theorem, using some previous results on analytic functions connected with open Riemann surfaces.The Cousin theorems, concerning the existence of analytic functions of several complex variables with prescribed poles and zeros in a given domain, have been generalized in various manners, but only in the case where the domain is schlicht. The authors proceed to the case where the given domain is the direct product of n open Riemann surfaces. They prove the following two theorems.

We consider one-parameter families of distributions which are of the general exponential type, the general series distribution type, or which can be transformed into the exponential type by a one-to-one transformation. In this paper we establish theorems to the effect that such distributions may be characterized by a simple differential equation involving the mean function. It is illustrated that almost all the classical one-parameter families of distributions are characterized by these theorems. Multivariate generalizations are given, and it is also noticed that the functional form of the normalizing factor determines the specific distribution in each general family.

Analogues of the central limit theorem and iterated logarithm law have recently been obtained for the Galton-Watson process; similar results are established in this paper for the temporally homogeneous Markov branching process and for the associated increasing process consisting of the number of splits in the original process up to time t. CENTRAL LIMIT THEOREM; ITERATED LOGARITHM LAW; BRANCHING PROCESS; TEMPORALLY HOMOGENEOUS MARKOV PROCESS; SPLITTING PROCESS

A von Neumann algebra is called hyperfinite if it is the weak closure of an increasing sequence of finite-dimensional von Neumann subalgebras. For a separable infinite-dimensional Hilbert space the following is known: there exist hyperfinite and non-hyperfinite factors of type II1 (4, Theorem 16’), and of type III (8, Theorem 1); all hyperfinite factors of type Hi are isomorphic (4, Theorem 14); there exist uncountably many non-isomorphic hyperfinite factors of type III (7, Theorem 4.8); there exist two nonisomorphic non-hyperfinite factors of type II1 (10), and of type III (11). In this paper we will show that on a separable infinite-dimensional Hilbert space there exist three non-isomorphic non-hyperfinite factors of type II1 (Theorem 2), and of type III (Theorem 3).Section 1 contains an exposition of crossed product, which is developed mainly for the construction of factors of type III in § 3.

In this paper we present three results involving function constructions, together with an example which shows the usefulness of our considerations. The first result, Theorem 1, states roughly that homotopy is self related through function space constructions. Section 1 is devoted to the precise statement and proof of this theorem.In order to clarify the results of § 2 we draw an analogy. Recall that under certain mild restrictions on the spaces involved, a map p: E → B is a fibration, and a map i: A → X is a cofibration, if and only if the induced maps p*: EZ → BZ and i*: ZX → ZA are fibrations for all spaces Z. The two theorems of § 2 give, in a more general category, analogous results for the notions of limit and colimit. Finally, in § 3 we suggest the importance of this type of result by using it, together with Theorem 1 and the above analogy, to deduce a result from its dual.

In this paper we study generalized homomorphisms between two algebras, namely the binary relations whose graphs are subalgebras of the direct product of the given algebras. In 1897 Goursat proved that every subgroup of the direct product of two groups is determined by an isomorphism between factor groups of subgroups of the given groups (10, §§11, 12; 25, pp. 15, 16). A like result is here shown for a general class of algebras, including loops and quasigroups, by a method due to Riguet (22). This result is used to obtain general forms of the Zassenhaus lemma and the Jordan- Hälder-Schreier theorem for normal series (26, §9).

Let Mn denote the space of all n-square matrices over an algebraically closed field F. For A, B ∊ Mn, letdefine the iterated commutators of A and B. Recently several research papers (1, 2, 4, and 5) have appeared on these commutators. In (1), Kato and Taussky have proved that for n = 2 the iterated commutators of A and B satisfy the linear relation

Various conditions under which an upper semi-continuous (u.s.-c.) decomposition of E3 yields E3 as its decomposition space have been given by Armentrout (1; 2; 5), Bing (7; 8), Lambert (13), McAuley (14), Smythe (17), and Wardwell (18). If the projection of the non-degenerate elements is 0-dimensional in the decomposition space, then “shrinking” or “Condition B” (6) has proven particularly useful.In this paper we shall investigate monotone u.s.-c. decompositions of a locally compact connected metric space M, where the projection of the nondegenerate elements is 0-dimensional. We show in Theorem 1 that each open covering of the non-degenerate elements of a 0-dimensional decomposition has a locally finite refinement.In § 5, we use Theorem 1 to investigate the following question which is similar to one raised by Bing (11, p. 19): Let G, G′, and G″ be decompositions of M such that the non-degenerate elements of G are those of G′ together with those of G′.

Let be a difference field of characteristic 0, m an irreducible manifold of effective order n over {y}, and F an algebraically irreducible difference polynomial in {y} of effective order n + k, k > 0, which vanishes on 3 m. In an earlier paper (2, p. 447) I gave necessary conditions, restated below as (a), (b), and (c) of the main theorem, for m to be an essential singular manifold of F. These conditions are analogous to the low power criterion of Ritt (1, p. 65) for the corresponding problem of differential algebra. Like that criterion they depend, in the special case that m is the manifold of y, only on which power products appear effectively in F. Unlike the low power criterion, however, conditions (a), (b), and (c) are only necessary, not sufficient.

Various problems concerning the general theory of centralizers of modules which are not assumed to be completely reducible have been discussed by Fitting (3), Brauer (2), and Nakayama. In this paper we present a new approach to some of these questions, which has its origin in Weyl's discussion (15) of the centralizer of a finite group of collineations.

If G is a torsion–free group and F is a field, is the group ring F[G] a ring without zero divisors? This is true if G is an ordered group or various generalizations thereof - beyond this the question remains untouched. This paper proves a related result.

In this paper, results obtained by the author for Riemannian Spaces Vn imbedded in Euclidean Spaces EN (3; 4) are extended to Vn imbedded in VN.The first section is introductory. In §2 the general result is obtained. This is the establishment of a certain dependency among the three basic sets of equations of the Vn with respect to the VN, namely the equations of Gauss, Codazzi and Kuehne.

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.

The theory of generalized Frattini subgroups of a finite group is continued in this paper. Several equivalent conditions are given for a proper normal subgroup H of a finite group G to be a generalized Frattini subgroup of G. One such condition on H is that K is nilpotent for each normal subgroup K of G such that K/H is nilpotent. From this result, it follows that the weakly hyper-central normal subgroups of a finite non-nilpotent group G are generalized Frattini subgroups of G.Let H be a generalized Frattini subgroup of G and let K be a subnormal subgroup of G which properly contains H. Then H is a generalized Frattini subgroup of K.Let ϕ(G) be the Frattini subgroup of G. Suppose that G/ϕ(G) is nonnilpotent, but every proper subgroup of G/ϕ(G) is nilpotent. Then ϕ(G) is the unique maximal generalized Frattini subgroup of G.

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).

Herein, a ring is not required to have an identity. All rings are associative but not necessarily commutative. However, we specialize to the commutative case for some of our results. The paper is concerned primarily with rings having the property that all unbounded ideals or all unbounded homomorphic images are isomorphic to the ring. We say that a ring R is bounded if nR = 0 for some positive integer n; alternately, R, with or without 1, is said to have finite characteristic. Unbounded rings having the property that all proper subrings are bounded were characterized in [8].

In 1946 Montgomery and Samelson (11) introduced a generalization of the notion of a differentiable group action with one type of orbit besides fixed points. Such an object is essentially a locally trivial fibering except on a certain singular set over which fibres are pinched to points. In recent years there has been a fair amount of research on these MS-fiberings and similar singular fiberings. This paper is another effort in this direction. For a fairly complete bibliography of the literature, the reader should consult the references, and in particular, (5).Let f: Mn → Sp, with Mn a closed connected n-manifold and Sp the unit p-sphere with standard differentiable structure, be the projection map of a smooth MS-fibering with finite non-empty singular set.

In the course of investigating the structure of finite groups which have a representation in the form ABA, for suitable subgroups A and B, we have been forced to study groups G which admit an automorphism ϕ such that every element of G lies in at least one of the orbits under ϕ of the elements g, gϕr(g), gϕrϕ(g)ϕ2r(g), gϕr(g)ϕr2r(g)ϕ3r(g), etc., where g is a fixed element of G and r is a fixed integer.In a previous paper on ABA-groups written jointly with I. N. Herstein (4), we have treated the special case r = 0 (in which case every element of G can be expressed in the form ϕi(gj)), and have shown that if the orders of ϕ and g are relatively prime, then G is either Abelian or the direct product of an Abelian group of odd order and the quaternion group of order 8.

In this paper we generalize to Riemann surfaces a theorem of Helson and Lowdenslager in (2) describing the closed subspaces of L2(﹛|z| = 1﹜) that are invariant under multiplication by eiθ.Let R be a region on a Riemann surface with boundary Γ consisting of a finite number of disjoint simple closed analytic curves such that R ⋃ Γ is compact and R lies on one side of Γ. Let dμ be the harmonic measure on Γ with respect to a fixed point t0 on R. We shall consider the closed subspaces of L2(Γ, dμ) that are invariant under multiplication by functions in A (R) = ﹛F|F continuous on , analytic on R}.

The aim of this paper is to characterize the finite simple group PSp4(3) by the structure of the centralizer of an involution contained in the centre of its Sylow 2-subgroup. More precisely, we shall prove the following result.

Our aim in this paper is to consolidate and extend some of the notions in (1; 2; 5; 6) concerning free planes in order to facilitate the study of their collineation groups. An upper bound mn for the orders of the finite subroups of Gn will be established, where Gn is the collineation group of the free plane ƒn of rank n + 4. In the process, a result of (6) will be generalized. Indeed, mn will be shown to be the best upper bound for all n ≠ 5.

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.

In [8] I. N. Herstein conjectured that all the finite odd order sub-groups of the multiplicative group in a division ring are cyclic. This conjecture was proved false in general by S. A. Amitsur in [1]. In his paper Amitsur classifies all finite groups which can appear as a multiplicative subgroup of a division ring. Let D be a division ring with prime field k and let G be a finite group isomorphic to a multiplicative subgroup of D.

Historically, the first example of a ring of quotients was the quotient field of an integral domain. Later on, conditions were found under which a noncommutative integral domain has a quotient division ring. More recently, R.E. Johnson [4], Y. Utumi [5], and G.D. Findlay and J. Lambek [3] have discussed the existence and structure of a maximal ring of quotients of any ring.The present paper uses the methods of Findlay and Lambek to recast the results of Johnson on the quotient ring of a ring with zero singular ideal. It is also shown that such a ring has a unique left-right maximal ring of quotients.

The determination of the abelian groups in which every neat subgroup is pure is a relatively routine exercise (see [6]). There are numerous problems of this type; for example, the determination of the groups in which every pure subgroup is isotype or the groups in which every subgroup is isotype. These are all special cases of the general problem of determining the abelian groups in which every α-pure subgroup is β-pure for arbitrary ordinal numbers α and β. The solution of this general problem is the object of this paper.

In this paper we investigate indefinite Finsler spaces in which the metric tensor has signature n — 2. These spaces are a generalization of Lorentz manifolds. Locally a partial ordering may be defined such that the reverse triangle inequality holds for this partial ordering. Consequently, the spaces we study may be made into what Busemann [3] terms locally timelike spaces. Furthermore, sufficient conditions are obtained for an indefinite Finsler space to be a doubly timelike surface (see [2; 4]). In particular, all two-dimensional pseudo-Riemannian spaces are shown to be doubly timelike surfaces.

In this paper we classify finite groups G with a faithful, quasiprimitive (see Notation), unimodular representation X with character χ of degree six over the complex number field. There are three gaps in the proof which are filled in by [16; 17]. These gaps concern existence and uniqueness of simple, projective, complex linear groups of order 604800, |LF(3, 4)|, and |PSL4(3)|. By [19], X is a tensor product of a 2-dimensional and a 3-dimensional group, or a subgroup thereof, or X corresponds to a projective representation of a simple group, possibly extended by some automorphisms. The tensor product case is discussed in section 10. Otherwise, we assume that G/Z(G) is simple. We discuss which automorphisms of G/Z(G) extend the representation X (that is, lift to the central extension G and fix the character corresponding to X) just after we find X(G).

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]).

In [6], H. Sharp gives a matrix characterization of each topology on a finite set X = {x1, x2,…, xn}. The study of quasi-uniform spaces provides a more natural and obviously equivalent characterization of finite topological spaces. With this alternate characterization, results of quasi-uniform theory can be used to obtain simple proofs of some of the major theorems of [1], [3] and [6]. Moreover, the class of finite topological spaces has a quasi-uniform property which is of interest in its own right. All facts concerning quasi-uniform spaces which are used in this paper can be found in [4].

The group ring AG of a group G and a ring A is the ring of all formal sums Σg∈G agg with ag ∈ A and with only finitely many non-zero ag. Elements of A are assumed to commute with the elements of G. In (2), Connell characterized or completed the characterization of Artinian, completely reducible and (von Neumann) regular group rings ((2) also contains many other basic results). In (3, Appendix 3) Connell used a theorem of Passman (6) to characterize semi-prime group rings. Following in the spirit of these investigations, this paper deals with the complete ring of (right) quotients Q(AG) of the group ring AG. It is hoped that the methods used and the results given may be useful in characterizing group rings with maximum condition on right annihilators and complements, at least in the semi-prime case.

A bisimple semigroup S is called I-bisimple if Es, the set of idempotents of S, with its natural order is order-isomorphic to I, the set of integers, under the reverse of the usual order. In (9), the author completely determined the structure of I-bisimple semigroups mod groups; in this paper, he also gave an isomorphism theorem, a homomorphism theorem, an explicit determination of the maximal group homomorphic image, and a complete determination of the congruences for these semigroups.

A well-known theorem, due to R. H. Bruck ([4], p. 398), is the following:If a finite projective plane of order n has a projective subplane of order m < n, then either n = m2 or n > m 2+ m.In this paper we prove an analagous theorem concerning affine subplanes of finite projective planes (Theorem 1). We then construct a number of examples; in particular we find all the finite Desarguesian projective planes containing affine subplanes of order 3 (Theorem 2).

This paper is concerned with the recent work of A. W. Goldie (1 ) on principal right ideal rings (p.r.i. rings). We shall prove some of his main structure theorems using the methods of (3) and (4), and in so doing shall weaken some of his hypotheses.

Let Σ be a left amenable semigroup, and let {Tσ: σ ∊ Σ} be a representation of Σ as a semigroup of positive linear contraction operators on L1(X, 𝓐, p). This paper is devoted to the study of existence of a finite equivalent invariant measure for such semigroups of operators.

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.

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)

In his fundamental paper [3], Hurwitz showed that the order of a group of biholomorphic transformations of a compact Riemann surface S into itself is bounded above by 84(g – 1) when S has genus g ≧ 2. This bound on the group of automorphisms (as we shall call the biholomorphic self-transformations) is attained for Klein's quartic curve of genus 3 [4] and, from this, Macbeath [7] deduced that the Hurwitz bound is attained for infinitely many values of g.After genus 3, the next smallest genus for which the bound is attained is the case g = 7. The equations of such a curve of genus 7 were determined by Macbeath [8] who also gave the equations of the transformations. The equations of these transformations were found by using the Lefschetz fixed point formula. If the number of fixed points of each element of a group of automorphisms is known, then the Lefschetz fixed point formula may be applied to deduce the character of the representation given by the group acting on the first homology group of the surface.

J. Lambek and G. Michler [3] have initiated the study of a ring of quotients RP associated with a two-sided prime ideal P in a right noetherian ring R. The ring RP is the quotient ring (in the sense of [1]) associated with the hereditary torsion class τ consisting of all right R-modules M for which HomR(M, ER(R/P)) = 0, where ER(X) is the injective hull of the R-module X.In the present paper, we shall study further the properties of the ring RP. The main results are Theorems 4.3 and 4.6. Theorem 4.3 gives necessary and sufficient conditions for the torsion class associated with P to have property (T), as well as some properties of RP when these conditions are indeed satisfied, while Theorem 4.6 gives necessary and sufficient conditions for R to satisfy the right Ore condition with respect to (P).

In this paper we consider complete orthonormal sequences defined on the interval [0, 1] and satisfying an inequality of the type(1)for allnand some sequence {Fn}. Such sequences were first considered by Zygmund and Marcinkiewicz (8). They extended the well-known results of Hausdorff-Young and Paley, originally proved for the casev= ∞,Fn=Mfor alln(12). We will consider cases of equality in t he Hausdorff-Young theorems and certain limiting cases of the Paley theorems. Application of these results and the results in (8) will be made to functions harmonic in the unit α-sphere.

Stone lattices were (named and) first studied in 1957 (5). Since then, a great number of papers have been written on Stone lattices and a very satisfactory theory evolved. Despite the fact that all chains with 0, 1 as well as all Boolean algebras are Stone lattices, it turns out that many of the nice theorems on Boolean algebras have analogues, in fact, generalizations for Stone lattices. To give just two examples: the characterization of Boolean algebras in terms of prime ideals (Nachbin (6)) is generalized in (5) (see also (9)); Stone's representation theory (8) is generalized in (4); see also (2).

Descriptive definitions of Cesàro-Denjoy integrals (CD-integrals) equivalent to the Cesàro-Perron integrals (CP-integrals) introduced by J. C. Burkill [1, 2] have been given by Miss Sargent [6] (see §2). The CD§integrals are generalizations of the special Denjoy integral [5, p. 201]. They are somewhat complicated in that modifications of the definitions of continuity, generalized absolute continuity in the restricted sense (ACG*) [5, p. 231], and of derivatives are required for each order. In the present paper a scale of integrals is obtained which is based on the descriptive definition of the general Denjoy integral [5, p. 241]. The approximate derivative and a slightly modified definition of generalized absolute continuity (ACG) are used for all orders so that the only concept generalized for increasing orders is that of continuity.

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.

This paper is an extension of the ideas discussed in (3, §§ 14-16); the extension consisting of the use of the third and fourth symmetric Riemann derivative instead of the Schwarz or second symmetric Riemann derivative.The J2integral, due to James (1), is defined in (3) as follows. Let f(x) be measurable on [a, b] and finite at each point; if there exists a continuous function F(x) such that D2F = f everywhere on (a, b),then1.1The definition is unique since if F(x) and G(x) are continuous and D2F =D2G everywhere then

In (4) Malcev generalizes the notion of the Lie algebra of a Lie group to that of an anti-commutative "tangent algebra" of an analytic loop. In this paper we shall discuss these concepts briefly and modify them to the situation where the cancellation laws in the loop are replaced by a unique two-sided inverse. Thus we shall have a set H with a binary operation xy defined on it having the algebraic properties(1.1) H contains a two-sided identity element e;(1.2) for every x ∊ H, there exists a unique element x-1 ∊ H such that xx-1 = x-1x = e

In this paper we prove that the order of the collineation group of the Veblen-Wedderburn plane of order nine is 311,040. This result was stated by Hall [3] in 1943 and proved by Pierce [9] in 1964. Hall assumed that there were 10 · 8 · 6 · 4 · 2 = 3840 collineations which permute points on the ideal line L and 81 collineations which leave L pointwise fixed. In 1955 André [1] verified this assumption. When it was realized that a harmonic homology with axis L had been overlooked, the number of central collineations with axis L doubled and hence the order of the collineation group became 3840 · 162 = 622,080. This latter figure has been assumed to be correct as recently as 1965 ([6]).Here it is proved that there are 1920 collineations which move points on L and 162 collineations which leave L pointwise fixed, thus giving the figure 311,040. Pierce's proof of this fact is established from a different viewpoint.

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.

It is well known that a supersoluble group has the nilpotent commutator subgroup. But the converse does not hold in general. Therefore, it will be interesting to study the class of groups with nilpotent commutator subgroups. Some results were obtained by B. Huppert [1] and R. Baer [3] on this subject.It is easily seen that the p-length of a group with the nilpotent commutator subgroup is 1 for every p dividing the order of this group. Theorem 1 of this paper may be considered as its local refinement, and is proved in a similar way as B. Huppert [2]. Theorem 2 gives a necessary and sufficient condition for a group to have the nilpotent commutator subgroup in terms of its Hall subgroups. In the last theorem 3 we shall study an interesting class of groups which have maximal subgroups of a certain nature. It turns out that these groups have nilpotent commutator subgroups and admit Sylowtowers.

1. Let k be an infinite field and let V/k be an irreducible variety of dimension ≧ 2 in a projective n-space Pn over k. Let P and Q be two k-rational points on V In this paper, we describe ideal-theoretically the generic hyperplane section of V through P and Q (Theorem 1) and prove that the section is almost always an absolutely irreducible variety over k1/pe if V/k is absolutely irreducible (Theorem 3). As an application (Theorem 4), we give a new simple proof of an important special case of the existence of a curve connecting two rational points of an absolutely irreducible variety [4], namely any two k-rational points on V/k can be connected by an irreducible curve.I wish to thank Professor A. Seidenberg for his continued advice and encouragement on my thesis research.

Throughout this paper will denote a C*-algebra and will denote its second dual, which is identified with the enveloping von Neumann algebra of . A projection E in is said to be open if it supports a left ideal in , that is, if E = - for some left ideal in . Here the bar - means the stong closure. When has a unit, this definition coincides with the definition of Akemann [1, Definition II. 1]. In the sequel, we shall solely be concerned with two-sided ideals, and consequently central projections [4, I, § 3, Corollary 3 of Theorem 2]. Our aim is to show that is CCR if and only if the open central projections are strongly dense in the set of central projections on .