Let G be a connected compact Lie group and H a connected closed subgroup. Then H is an orientable submanifold of G and we may consider H as a cycle in G. In his interesting paper on the topology of group manifolds H. Samelson has proved that, if H is not homologous to 0, then the homology ring of the coset space G/H is isomorphic to the homology ring of a product space of odd dimensional spheres and the homology ring of G is isomorphic to that of the product of the spaces H and G/H. On the other hand, in a recent investigation of fibre bundles’ T. Kudo has shown that, if the homology ring of the coset space G/H is isomorphic to that of an odd dimensional sphere, then H is not homologous to 0.

The theory of group characters was originated by G. Frobenius and has been studied by many authors including, above all, I. Schur and W. Burnside. As to the modular theory we owe it to recent works by R. Brauer and his collaborators, including T. Nakayama and C. Nesbitt, which clarified also the connections between the structure theory and the representation theory.

Let G be a locally compact group. Under a neighbourhood U we mean a symmetric (i.e. U = U−1) neighbourhood of the identity e, with the compact closure Ū. If there exists a neighbourhood U containing no subgroup other than the identity group, we say that G has no small subgroup.

Der. Umschwung, welchen die mathematischen Grundlagenforschungen in diesem Jahrhundert erfuhren, teilte früher einmal die Untersuchungen in drei Hauptrichtungen ab, nämlich in die intuitionistische, logistische und formalistische Auffassung der Mathematik, wonach die lebhafte Polemik über das Wesen der Mathematik sowie das Verhältnis der Logik zur Mathematik aufblütete, und zwar am heftigsten in den zwanziger Jahren dieses Jahrhunderts. Nicht in den prinzipiellen Einsichten, sondern in der Methode der Forschungen hatten doch diese drei Standpunkte viele Berührungspunkte. Erstens entlehnt der Hilbertsche Formalismus dem Logizismus das Hilfsmittel zur Formalisierung der Mathematik. Zweitens ragt die Beweisführung in der Metamathematik, welche die formalisierte Mathematik als ihren eigenen Gegenstand der Untersuchungen betrachtet, insofern nicht aus der intuitionistischen Mathematik hervor, als man die formalisierte Mathematik nur als Zeichenkombinationen ansieht. Ferner sind die formalen Regeln, welche in der intuitionistischen Denktätigkeit erfindlich sind, zuerst von Herrn Heyting formalisiert, danach von Herren Gödel, Gentzen u. a, der logistischen Untersuchungen unterworfen worden. Überdies haben diese drei Richtungen der Grundlagenforschungen heutzutage, wozu gemeinsam der erste Anstoss durch die Cantorsche Mengenlehre gegeben worden ist, auch ihre Absicht in Gemeinschaft, welche dar in besteht, die Logik in bezug auf das Unendliche zu verdeutlichen und damit der Mathematik sichere Stellung zu liefern.

In my previous paper “On the theory of semi-local rings,” we saw that if a semi-local ring R with maximal ideals p1…,ph is a subdirect sum of local rings Rm,2) then R is the direct sum of R[p1] (proposition 15, (slr)1)) and that a complete semi-local ring is a direct sum of complete local rings (Remark to proposition 5, (slr)).

A curve c2 of finite length L2 moves on a euclidean plane. Let the number of points of intersection of c2 with the fixed, curve C1 of length Ls1 be n, and the element of kinematic measure of the position of c2 be dK.

In this note it will be proved that some kinds of Lie groups (including semi-simple Lie groups) have an everywhere dense subgroup which is algebraically isomorphic to the free group generated by two elements (Theorem 8),

In § 1 characterizations of Lie groups which are approximated by discrete subgroups’ are given. This section is closely connected with a part of the results of Malcev [4] and Matsushima [5], and some theorems are slight modifications of them (Theorems 2 and 3).

It is well known that the fundamental group π1(X) of an arcwise connected topological space X operates on the n-th homotopy group πn(X) of X as a group of automorphisms. In this paper I intend to construct geometrically a group 𝒰(X) of automorphisms of πn(X), for every integer n ≥ 1, which includes a normal subgroup isomorphic to π1(X) so that the factor group of 𝒰(X) by π1(X) is completely determined by some invariant Σ(X) of the space X. The complete analysis of the operation of the group on πn(X) is given in §3, §4, and §5,

In K, Itô’s paper [1] on the theory of turbulence, the problem to determine the canonical form of the moment tensor of temporally homogeneous and isotropic turbulence, has not been solved. In the present paper, the author will solve the problem by making use of the result of his preceding paper [2]. We shall treat the turbulence in R3 but the similar argument is possible in Rn.

A. Weil proposed in his book “Foundations of algebraic geometry” several problems concerning differential forms on algebraic varieties, S. Koizumi has proved that if ω is a differential form on a complete variety U without multiple point, which is finite at every point of IT, then ω is the differential form of the first kind. The following example shows that on everywhere normal varieties with multiple points this statement holds no more; that is: A differential form on a everywhere normal variety which is finite on every simple point of its variety is not always the differential form of the first kind.

Recently A, Malcev has shown that the homogeneous space of a connected nilpotent Lie group G is the direct product of a compact space and an Euclidean-space and that the compact space of this direct decomposition is also a homogeneous space of a connected subgroup of G. Any compact homogeneous space M of a connected nilpotent Lie group is of the form where is a connected simply connected nilpotent group whose structure constants are rational numbers in a suitable coordinate system and D is a discrete subgroup of G.

A conjecture concerning characterization of blocks of representations of symmetric groups by means of Young diagrams, given formerly by one of the writers, was affirmed recently by R. Brauer and G. de B. Robinson jointly. Namely, it was proved in Brauer’s paper, relying on his general, and profound, theory of blocks in finite groups, that the number of p-blocks in a symmetric group of degree n, is exactly equal to the number of p-cores, or p-kernels in the terminology of NI, with n- Ip (l= 0, 1, 2,…) nodes, and that certain statements designated as (A), (B) and (C) would secure that two irreducible representations with a same p-core belong to a same block (whence the conjecture itself when combined with the estimate of the number of blocks). Then these statements (A), (B), (C) were proved actually in Robinson’s paper.

Let A be a primary algebra with unit element over a field K and Z its center. Let Ā be the simple residue class algebra of A modulo its radical. Then it is known, and can readily be seen, that there holds the inequality where t is the rank of A over its center. We call A maximally central if in particular i.e. if the rank [Z: K] takes its maximum value. Further, an algebra which is a direct sum of those primary algebras will be called maximally central too. The notion was introduced in Azumaya-Nakayama [5], as a by-product of the study of absolutely uni-serial algebras.