The principal aim of this note is the proof of the following:

Main Theorem. The following properties of the group G are equivalent:

I. G is artinian and soluble.

II. (a) To every maximal subgroup S of G there exists a normal subgroup T of G with G = ST and S ∩ T = (S ∩ T)T.

In (4) G. Lewittes proved some theorems connecting automorphisms of a compact Riemann surface with the Weierstrass points of the surface, and in (5) he applied these results to elliptic modular functions. We refer the reader to these papers for definitions and details. It is our purpose in this note to point out that these results are of a purely algebraic nature, valid in arbitrary algebraic function fields of one variable over algebraically closed ground fields (with an obvious restriction on the characteristic). We shall also make use of the calculation carried out in (5) to obtain a rather easy extension of a theorem proved in (6, p. 312).

H. Liebmann (3) and W. Süss (7) proved

Theorem A. The only convex closed hypersurface with constant mean curvature in a Euclidean space is a sphere.

Y. Katsurada (1; 2) gave the following generalization.

Theorem B. Let M be an orientable Einstein space which admits a proper conformai Killing vector field, that is, a vector field generating a local one-parameter group of conformai transformations which is not that of isometries, and S a closed orientable hypersurface in M whose first mean curvature is constant. If the inner product of the conformai Killing vector field and the normal to the hypersurface has fixed sign on S, then every point of S is umbilical.

La caractérisation fondamentale des sous-groupes du groupe additif R des nombres réels en tant que groupes ordonnés est la suivante:

A. Un groupe abélien totalement ordonné G est isomorphe à un sous-groupe de R si et seulement si G est archimédien.

Nous nous proposons ici de donner une nouvelle caractérisation des sousgroupes de R qui nous semble présenter un certain intérêt pour plusieurs raisons. D'abord elle est très différente de la caractérisation A ci-dessus, bien que notre démonstration utilise A. Puis elle ajoute un nouveau théorème de r-groupe à ceux déjà connus. Finalement, en employant les caractérisations des anneaux de valuation généraux et discrets données dans (1), le théorème en question possède quelques corollaires parmi lesquels se trouve le théorème de s-groupe pour un domaine d'intégrité (“s-Gruppensatz” dans la terminologie de Krull (5)).

The present paper began as a natural outgrowth of our first paper, where we characterized the module homomorphisms from group algebras into a fairly restrictive class of group algebra modules. We now investigate module homomorphisms from group algebras into a more general class of group algebra modules. Although the two papers are thus related, they can be read quite independently.

Section 2 contains our extension, Theorem 2.1, of P. J. Cohen's theorem on factorization in Banach algebras (1). Our extension is to Banach modules over Banach algebras equipped with an approximate identity. We should mention first that J.-K. Wang observed the existence of such a generalization, and secondly, that our proof requires no ideas different from those in Cohen's proof. Nevertheless, we include a proof that condenses the original proof considerably.

In der ebenen absoluten Geometrie, die hier stets in der Allgemeinheit meines Buches Aufbau der Geometrie aus dem Spiegelungsbegriff (AGS) verstanden sei, hat man guten Grund, die Geraden vor den Punkten zu bevorzugen. Grundlegende Relationen in der Menge der Geraden sind das Senkrechtstehen von zwei Geraden und das Im-Büschel-Liegen—die Abhängigkeit—von drei Geraden, das man nach Hjelmslev wie folgt definiert: Drei Geraden liegen im Büschel, wenn das Produkt der Spiegelungen an ihnen eine Geraden-Spiegelung ist.

Let a semigroup A be given by generators a1, a2, … , ad and defining relations u1 = v1, u2 = v2, … , ue = ve between these generators, the ui, vi being words in the generators. We then have a presentation of A, and write

The same generators with the same relations can also be interpreted as the presentation of a group, for which we write

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.

We shall be concerned at first with some properties of the solutions of the matrix differential equation

1.1

where

is an n × n symmetric matrix whose elements are continuous real-valued functions for 0 < x < ∞, and Y(x) = (yij(x)), Y″(x) = (y″ ij(x)) are n × n matrices. It is clear such equations possess solutions for 0 < x < ∞, since one can reduce them to a first-order system and then apply known existence theorems (6, Chapter 1).

In this paper we consider the following problem. Let (E, M, N, π) be a differentiable fibre bundle, where E is the total space, M the base space, N the fibre, and π: E → M the projection map. Then, given a Cr triangulation (ƒ, D) of M, can one obtain a Cr triangulation (F, K) of E such that the induced map ƒ–1πF: K → D is linear? R. Lashof and M. Rothenberg (3) have obtained this result for vector bundles.

Using methods quite different from theirs, we obtain a solution in the general case. The methods we use are the geometric methods developed by J. H. C. Whitehead. (7) in his triangulation of differentiable manifolds, as found in (5). In fact, our solution consists of generalizing his techniques in a fibre bundle setting.

Throughout this paper R and Rn will denote v-rings, that is, complete discrete rank-one valuation rings of characteristic zero, having a common residue field k of characteristic p. R is assumed unramified and Rn has ramification index n. Let π be a prime element in Rn. Then Rn = R[π], where π is a root of an Eisenstein polynomial ƒ = xn + pƒn-1 xn-1 + … + pƒ0 with coefficients in R and ƒ0 a unit. Thus Rn is inertially isomorphic to R[[x]]/ƒR[(x)], that is, the rings are isomorphic by a mapping which induces the identity mapping on the common residue field. R[[x]] represents the power series ring in the indeterminate x over R. In this paper we identify Rn with R[[x]]/ƒR[[x]], R with its natural embedding in Rn and π with x + ƒR[[x]].

For various classes Q of metric spaces, there are several well-known results characterizing the local n-connectivity of a metric space in terms of n-ANR(Q)'s. Specifically, we have in mind the results of Kuratowski (13, p. 265) and Kodama (10, p. 79). The main purpose of this paper will be to obtain similar results along these lines for non-metric classes Q. In the last part of the paper we specify Q to be the class of totally normal spaces and characterize the local n-connectivity of an n-dimensional separable metric space in terms of ANR(Q)'s.

Let ∑ be a projective space PG(3, q) of dimension 3 and finite order q. Then ∑ contains (q + 1)(q2 + 1) points and an equal number of planes, and (q2 + 1) (q2 + q + 1) lines. It will be convenient to consider lines and planes as sets of points and to treat the incidence relation as set inclusion. Each plane contains q2 + q + 1 points and an equal number of lines. Each line contains q + 1 points and is contained in an equal number of planes. Each point is contained in q2 + q + 1 planes and an equal number of lines.

A spread of lines of ∑ is a set of q2 + 1 lines of ∑ which are pairwise disjoint, or skew; it can also be defined as a set of lines such that each point (or each plane) is incident with exactly one of the lines.

Many authors have studied lattice congruences on lattices, but it seems that there are few studies concerning semilattice congruences on lattices. However, it seems that the semilattice congruences on lattices are closely connected with their structure. In this paper, we shall study the characterizations of modular, distributive, and relatively complemented lattices by the permutability of semilattice congruences.

We can obtain the dual statements of the following discussion, but we shall not write them as a rule to avoid double descriptions.

Given a probability measure space (Ω, , P) consider the following packing problem. What is the maximum number, b(K, Λ), of sets which may be chosen from so that each set has measure K and no two sets have intersection of measure larger than Λ < K?

In this paper the packing problem is solved for any non-atomic probability measure space. Rather than obtaining the solution explicitly, however, it is convenient to solve the following minimal paving problem. In a non-atomic a-finite measure space (Ω, , μ) what is the measure, V(b, K, Λ), of the smallest set which is the union of exactly b subsets of measure K such that no subsets have intersection of measure larger than Λ?

Kummer's surface has the base surface F of a certain net of quadrics in [5] for a non-singular model. All the quadrics of have a common self-polar simplex ∑, and can, in a double-infinity of ways, be based on a quadric Ω1 and two quadrics that Ω1 reciprocates into each other. F is invariant under harmonic inversions in the vertices and opposite bounding primes of ∑ and (§2) contains 32 lines. In §3 it is shown, conversely, that those quadrics for which a given simplex is self-polar and which contain a line of general position constitute a net of this kind.

In a recent paper (2), Helson and Kahane consider the problem of the existence of real numbers x such that the sequence (λn x) (when reduced modulo 1) is not summable by a given regular Toeplitz method, where (λn) is a lacunary sequence of positive real numbers. Thus, as an example, they show the existence of uncountably many x such that the sequence (θn x) does not have a distribution function modulo 1, where θ is some fixed number > 1.

A pro-p-group G is said to be a Demushkin group if

1. (1) dimFp H1(G, Z/pZ) < ∞,

2. (2) dimFp H2(G, Z/pZ) = 1,

3. (3) the cup product H1(G, Z/pZ) × H1(G, Z/pZ) → H2(G, Z/pZ) is a non-degenerate bilinear form. Here FP denotes the field with p elements. If G is a Demushkin group, then G is a finitely generated topological group with n(G) = dim H1(G, Z/pZ) as the minimal number of topological generators; cf. §1.3. Condition (2) means that there is only one relation among a minimal system of generators for G; that is, G is isomorphic to a quotient F/(r), where F is a free pro-p-group of rank n = n(G) and (r) is the closed normal subgroup of F generated by an element r ∈ F9 (F, F); cf. §1.4.

Let P be a projective plane of finite order n and Γ a group of collineations of P. Gleason (6) and Wagner (10) have shown that if every point of P is the centre, and every line the axis, of a non-trivial perspectivity in Γ, then Γ contains a subgroup of order n2 which consists entirely of elations. It then follows that either P or its dual is a translation plane with respect to at least one line; in fact if Γ has no fixed elements, then P is desarguesian and Γ contains all elations of P. It was shown by Piper (7) and Cofman (4) that the hypotheses of Gleason and Wagner can be relaxed in certain cases, while the same conclusions hold.

A closed curve Kn+1 of order n + 1 in real projective n-space Rn has a maximum number of n + 1 points in common with any (n — 1)-space. These curves are subjected to certain differentiability assumptions which make it possible to describe their singular points and to provide them with multiplicities in analogy with algebraic geometry.