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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.

The Plancherel's theorem which was established on an abelian LC (locally compact) group by M. Krein, D. Raikov, H. Cartan and R. Godement, etc., has been developed in recent years to the case that groups are unimodular LC; F. I. Mautner and I. E. Segal have constructed their theories under a little restriction, that is, the separability condition on account of their method of using von Neumann's reduction theory. More recently, R. Godement and otherwise H. Sunouchi have constructed their theories excepting such restriction and availing themselves of Radon's measure to the full....

Let(1) be a word in two variables X, Y, i.e. an element in the free group F2 on two generators X, Y. Let us say that f defines an associative composition for a group G if for arbitrary elements a, b, c in G we have(2) where a ° b is defined by(3)

Recently I. Gelfand and M. Neumark [2] have determined the types of irreducible unitary representations of the group G1 of linear transformations of the straight line. The analogous result is obtained for the group G2 of transformations z → az + b in the complex-number plane , where a and b run over all complex numbers with the exception of a = 0, which may be considered as the group of all sense-preserving similar transformations in the two-dimensional euclidean space E2.

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).

Let G be a group. If any finite number of elements from G generate a free subgroup in G, then G is called to be locally free. Some properties of locally free groups were given by A. Kurosch....

The present paper is a continuation of Part I. In the introduction of Part I we have explained our main problems and sketched their results. In the present part we shall proceed to give complete proofs of them.

Orderd groups, or o-groups, have been studied by G. Birkhoff, A. H. Clifford, H. Cartan, T. Nakayama, C. J. Everett and S. Ulam, J. von Neuman and others, while lattice ordered groups, or l-groups, also discussed by many mathematicians, G. Birkhoff and others. The present work is to establish the structure or order-buds (cf. below) in groups....

In connection with the class field theory a problem concerning p-groups was proposed by W. Magnus: Is there any infinite tower of p-groups G1, G2,…, Gn, Gn+1,…such that G1 is abelian and each Gn is isomorphic to Gn+1/θn(Gn+1), θn(Gn+1) ≠ 1, n = 1,2,…, where θn(Gn+1) denotes the n-th commutator subgroup of Gn+1? The present note is, firstly, to construct indeed such a tower, to settle the problem, and also to refine an inequality for p-groups of P. Hall.

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.

In this note, we give a characterization of inner automorphisms in the set of automorphisms of a compact connected group, and then apply it to give a proof of a theorem due to K. Iwasawa on the group of automorphisms of a compact group.

A formulation of the differentiability of the product function of a locally compact group at the identity may be given in the following way: In a sufficiently small euclidean neighborhood of the identity product satisfies the condition: where M(λ) is a function such that M(λ) → 0 as λ → 0, From this we can deduce easily.

Es ist wohlbekannt, dass die Frattini-Gruppe einer endlichen Gruppe G, mit anderen Worten, der Durchschnitt aller maximalen Untergruppen von G, nilpotent ist. Offen ist dagegen bis jetzt die Frage, ob die Faktorgruppe G/F*(G) von G nach dem Erzeugnis F*(G) aller minimalen Untergruppen eine besondere Struktur besitzt, z. B. ob sie auflösbar ist. Mehrere Schwierigkeiten sind der Grund hierfür: Einmal ist G/F*(G) im allgemeinen nicht nilpotent, zum anderen kennen wir die Struktur der einfachen und auflösbaren Gruppen (abgesehen von der Theorie von P. Hall) zu wenig. In der vorliegenden Note sollen teilweise Antworten auf die Frage nach der Auflösbarkeit von G/F*(G) gegeben werden.

In this note we shall apply the subgroup-theorem in free products of groups to a special class of subgroups, called word-subgroups, and study the free product decompositions of such subgroups. The commutator subgroup and related subgroups are especially studied....

Let G be a locally compact connected group, and let A (G) be the group of all continuous automorphisms of G. We shall introduce a natural topology into A(G) as previously (i.e. the topology of uniform convergence in the wider sense.) When the component of the identity of A(G) coincides with the group of inner automorphisms, we shall call G complete. The purpose of this note is to prove the following theorem and give some applications of it.

Let us consider soluble groups whose Sylow subgroups are all abelian. Such groups we callA-groups, following P. Hall.A-groups were investigated thoroughly by P. Hall and D. R. Taunt from the view point of the structure theory. In this note, we want to give some remarks concerning representation theoretical properties ofA-groups.

Es sei G eine im Kleinen kompacte Gruppe mit einem links-invarianten Haarschen Masse dx = m(dx) und es sei ferner L^1^,^p(G) der Durchschnitt zweier Banachschen Räume auf G, L^1(G) und L^p(G) für p > 1; offenbar bildet L^1^,^p(G) einen normietrten Ring mit der Norm...

Each of the five papers in the Journal of the Institute of Polytechnics, Osaka City University, Series A, Vol. 5, No. 2 have incorrect page headings, Vol. 4, No. 2 and should be corrected as Vol. 5, No. 2