Let G be a group. The Fitting subgroup F(G) of G is defined to be the set union of all normal nilpotent subgroups of G. Since the product of two normal nilpotent subgroups is again a normal nilpotent subgroup (see [10] p. 238), F(G) is the unique maximal normal, locally nilpotent sungroup of G. In particular, is G is finite, then F(G) is the unique maximal normal nilpotent subgroup of G. If G is a notrivial solvable group, then clearly F(G) ≠1.

It is customary to define a vector space in some such manner as: A vector space over a field F is a set V of elements a, b, … called vectors, having the following properties: For arbitrary a, b, c ∈ V; λ, μ ∈ F.

A method of deriving a class of solutions of Einstein's field equations with symmetry of de Sitter type is investigated, and the most general solution of this kind is derived.

In this paper a number of questions about locally compact groups are studied. The structure of finite dimensional connected locally compact groups is investigated, and a fairly simple representation of such groups is obtained. Using this it is proved that finite dimensional arcwise connected locally compact groups are Lie groups, and that in general arcwise connected locally compact groups are locally connected. Semi-simple locally compact groups are then investigated, and it is shown that under suitable restrictions these satisfy many of the properties of semi-simple Lie groups. For example, a factor group of a semi-simple locally compact group is semi-simple. A result of Zassenhaus, Auslander and Wang is reformulated, and in this new formulation it is shown to be true under more general conditions. This fact is used in the study of (C)-groups in the sense of K. Iwasawa.

Let Xi, ι = 1, 2, 3,… be a sequence of independent and identically distributed random variables and write S0 = 0, Sn = ∑ni=1Xi, n ≧ 1. Nn is the number of positive terms in the sequence S0, S1, S2,…, Snn ≧ 0.

In general, information concerning the distribution of the time to absorption, T, of a simple branching (Galton-Watson) process for which extinction in finite mean time is certain, is difficult to obtain. The process of greatest biological interest is that for which the offspring distribution is Poisson, having p.g.f. F(s) = em(s-1), m < 1.

Let k be a polygonal knot in Euclidean 3-space, p a projection onto a plane. If p;/k is 1:1 except at a finite number of points, which are not vertices of k and at which p/k is 2:1, then p(k) is said to be a regular projection of k this means that p(k) is closed curve with a finite number of double points (“crossings”) which are not points of tangency. Clearly for every polygonal knot there is a plane onto which it can be projected regularly. At each crossing of p(k), the knot k assigns an overcrossing arc and an undercrossing arc of the projection; conversely, if at each crossing we say which arc is an overcrossing, then there is a knot, uniquely determined up to homeomorphism, with this regular projection with the assigned overcrossings.

The object of this paper is to apply, in the case n = 4, the results of Barnes and Dickson [1] concerning extreme coverings of n-dimensional Euclidean space by equal spheres whose centres form a lattice.

In this note we shall deal with the enumeration of labelled trees of given order and given height over a selected point.

We consider infinite sequences of positive integers having exponential growth: and becoming ultimately periodic modulo each member of a rather sparse infinite set of integers. If sufficient, natural conditions are placed on the growth and periodicities of , we find that a is an algebraic integer having all its algebraic conjugates within or on the unit circle, and fn has a special representation involving an. The result is a kind of dual to the theorem of Pisot (cf. Salem [2], p. 4, Theorem A).

It is known that the Unified Field Theories of Weyl [14] and of Einstein [4] give no indication of how Relativity and Quantum Theories should be connected into a comprehensive field theory of physics. Indeed, the only determined attempt to establish such a theory, due to Eddington [3] and [6], failed through lack of contact with the contemporary developments, especially in quantum electrodynamics and elementary particles. Its author tried to explain curvature of the space-time in terms of statistical fluctuations partly of a physical origin defined within a mechanical system, and partly of a geometrical origin of coordinates superimposed on the latter. It is clear however that both the fluctuations of Eddington refer to purely mathematical frames. The probabilistic nature of his theory takes no account of physical objects, such as particles or energy distributions. It is the author's belief that this is the cause of difficulties associated with the otherwise admirable work of Eddington.

Let Fx and Fy denote families of subsets of the nonempty sets X and Y respectively and let be a function mapping Y onto X with the property that [H] ∈ Fx for each H∈Fy. Then the family l of all functions mapping X into Y such that [H] ∈Fy for each H ∈Fx is a semigroup if the product g of two such functions and g is defined by g = o o g (i.e., (Fg)(x) = (f(g(x))) for each x in X). With some restrictions on the families Fx and Fy and the function , these are the semigroups mentioned in the title and are the objects of investigation in this paper. The restrictions on fxfy and are sufficiently mild so that the semigroups considered here include such semigroups of functions on topological spaces as semi- groups of closed functions, semigroups of connected functions, etc.

Let (x) be a continuous function with period 2π. It is well known that the Fourier series of (x) is summable Riesz of any positive order to (x). The aim of this paper is the proof of the following theorem.

In his paper [8], N. Itô gives an elegant proof that the Sylow p-group of a finite solvable linear group of degree n over the field of complex numbers is necessarily normal if p > n+1. Moreover he shows that this bound on p is the best possible when p is a Fermat prime (i.e. a prime of the form 2sk + 1) but that the bound may be improved to p > n when p is not a Fermat prime.

A ring K is a radical extension of a subring B if for each x ∈ K there is aninteger n = n(x) > 0 such that xn ∈ B. In [2] and [3], C. Faith considered radical extensions in connection with commutativity questions, as well as the generation of rings. In this paper additional commutativity theorems are established, and rings with right minimum condition are examined. The main tool is Theorem 1.1 which relates the Jacobson radical of K to that of B, and is of independent interest in itself. The author is indebted to the referee for his helpful suggestions, in particular for the strengthening of Theorem 2.1.

In this paper a certain group with the third-Engel condition, that is a member of the variety defined by1 will be presented. Reasons for which its properties may be of interest are advanced in the present section.

The class of finite groups having a subgroup of order 4 which is its own centralizer has been studied by Suzuki [9], Gorenstein and Walter [6], and the present author [11]. The main purpose of this paper is to strengthen Theorem 5 of [11] by using an early result of Zassenhaus [12]. In particular, we find all groups of the class which are core-free, i.e. which have no nontrivial normal subgroup of odd order. As an application, we make a determination of a certain class of primitive permutation groups.