The present day theory of finite groups might be regarded as the outgrowth of the algebraic theory of equations. In much the same way one might consider the modern theory of infinite groups as stemming from late nineteenth century topology. The groups that crop up in topology are of a particularly simple type in that they are both finitely generated and finitely related. This means that every element in such a group can be expressed in terms of a finite number of elements and their inverses and every relation is an algebraic consequence of a finite number of relations between these elements. In other words the legacy of topology to group theory is the estate of finitely presented groups. This talk is concerned with the seemingly simplest of the finitely presented groups, the so-called groups with a single defining relator.

If K is a field and L an extension field, α ∈ L is said to be algebraic over K if it satisfies an equation with coefficients in K. If α does not satisfy any such on, it is called transcendental over K.

We examine the equation where with .

D. A. Edwards has shown [1] that if X is a locally compact Abelian group and f ∈ L∞, then the translate fa of f varies continuously with α if and only if f is (equal l.a.e. to) a bounded, uniformly continuous function. He remarks that this is a sort of dual to part of a result due to Plessner and Raikov which asserts that an element μ of the space Mb of bounded Radon measures on X belongs to L1 (i.e., is absolutely continuous relative to Haar measure) if and only its translates vary continuously with the group element, the relevant topology on Mb being that defined by the natural norm of Mb as the dual of the space of continuous functions vanishing at infinity. The proof he uses (ascribed to Reiter) applies equally well in both cases, and also to the case in which X is non-Abelian. A brief examination shows that in the latter case it is ultimately immaterial whether left- or right-translates are considered; since the extra complexities of this case are principally terminological, we shall direct no further attention to it.

If f(z) is analytic at the origin, f(0)=0, and f′(0)=λ, where 0< λ <1, then Koenigs' [3] solution of Schroeder's equation w(f(z))=λw(z), with multiplier λ, is given by . Here fn(z) denotes the nth iterate of f(z), defined inductively as f0(z)=z, f(fn-1(z)), n=1, 2, 3, …. More generally the solution w(z) of Schroeder's equation is uniquely determined to within a multiplicative constant by the requirement that it be analytic at the origin. From the uniqueness it follows that if g(z) is analytic at the origin, vanishes there, and commutes with f(z), i.e., f(g(z)) = g(f(z)), then w(g(z)) = w(z), for some multiplier a. Since w′(0) = 1 for Koenig's' solution, it has an inverse locally, and we find that g(z) is uniquely determined by its linear part; in fact g(z) = w-1(μw(z)). In particular the integral iterates of f can be put in the form w-1(λw(z)) for integral n. Thus for any α, real or complex, we may define fα(z), consistent with the above definition when α is a positive integer, as w-1(λw(z)). In this manner any function g(z) of the above type can be considered as an iterate of /(z). Also if cc, j9, 0 are any two distinct points sufficiently close to the origin there exists an analytic function g(z) which commutes with f(z) such that g(0) = 0, g(α) = β. In fact g(z)=w-1(χw(z)), where the multiplier χ=w(β)(w(α))-1. These facts are all well known, e.g. [2] [5], and we shall establish analogous results in a more general situation.

Let f(x) and g(x) be two polynomials with arbitrary complex coefficients that are relatively prime. Hence the maximum is positive for all complex x. Since m(x) is continuous and tends to infinity with |x|, the quantity E(f, g) = min m(x) is therefore also positive.

In connection with Relativity, Kottler [2] introduced the space V4 whose metric tensor is given by xi being space coordinates and Ф being a function of x1 only, and showed that if Where a and b are arbitrary constants, then the V4 is an Einstein space.

In a paper of nearly thirty years ago (Mahler 1937) I first studied approximation properties of algebraic number fields relative to their full system of inequivalent valuations. I now return to these questions with a slightly improved method and establish a number of existence theorems for such fields.

Let G be a finite group all of whose proper subgroups are nilpotent. Then by a theorem of Schmidt-Iwasawa the group G is soluble. But what can we say about a finite group G is only one maximal subgroup is nilpotent? Let G be a finite group with a nilpotent maximal subgroup M.

The 3-metabelian groups are those groups in which every subgroup generated by three elements is metabelian. In [2] it was stated and it [3] it was proved that the variety of such groups may be defined by the one law (by [a, b, c, d] we mean [[a, b], [c, d]], and for other definitions and notation we refer to [1]). Recently Bachmuth and Lewin obtained in [1] the surprising and remarkable result that the same variety is defined by the law Now (2) is reminiscent of the relation which holds in all groups and which is apparently due to Philip Hall. Using the identities , etc., we find that (2) is equivalent to where u = [z, x, y] and v = [y, z, z] [z, x, y]. Note that apart from certain displeasing conjugates (4) is curiously similar to both (1) and (2).

Let (G) denote the lattice of all subgroups of a group G. By an -isomorphism (lattice isomorphism) of G onto a group H, we mean an isomorphism of (G) onto (H). By an -isomorphism (normaliser preserving -isomorphism) of G onto H, we mean an -isomorphism ø such that (Aø) = (A)ø for all A ∈ (G). In this paper, we study certain properties of groups which remain invariant under -isomorphisms.

Let L be a finite dimensional Lie algebra over the Field F. We denote by (L) the lattice of all subalgebras of L. By a lattice isomorphiusm (whicn we abbrevite to -isomorphism) of L onto a Lie algebra M over the same field F, we mean an isomorphism of (L) onto (M). It is possible for non-isomorphic Lie algebras to -isomorphic, for example, the algebra of real vectors with product the vector product is -isomorphic to any 2-dimensional Lie algebra over the field of real numbers.

The functions where ξ > 0, are essentially Fourier convolutions of Gaussian and Cauchy type distributions, and have thus found application in many fields. In reactor theory, and no doubt elsewhere, the need has arisen to analyse complicated integrals and functional equations involving ψ and φ, the so-called Voigt profiles.

The paper considers the queueing system GI/G/1 with a type of customer impatience, namely, that the total queueing-time is uniformly limited. Using Lindiley's approach [10], an integral equation for the limiting waiting- time distribution is derived, and this is solved explicitly for M/G/1 using an expansion of the Pollaczek-Khintchine formula. It is also solved, in principle for Ej/G/l, and explicitly for Ej/Ek/l. A duality noted between GIA(x)/GB(x)/l and GIB(x)/GA(x)/l relates solutions for GI/Ek/l to Ek/G/l. Finally the equation for the busy period in GI/G/l is derived and related to the no-customer-loss distribution and dual distributions.

Kendall [4] has given for the distribution of the time to first emptiness in a store with an input process which is homogeneous and has non-negative independent increments and an output of one unit per unit time the formula . In this formula, z is the initial content of the store, g(t, z) is the density function of the time to first emptiness τ(z), defined by and k(t, x) is the density function of the input process ξ(t), defined by .

A great deal of attention has been given in the literature to the various properties of the simple binomial random walk. Explicit expressions are available for first passage times, absorption probabilities, average duration of the walk up to absorption and other quantities of interest. One aspect of the behaviour of this work which has, however, attracted little attention is the form of the distribution of occupation totals. This paper is devoted to the derivation of an explict expression for the joint probility generating function of the occupation totals up to absorption, for the binomial random walk in the presence of two absorbing points. The appropriate marginal form of this p.g.f. yields the distribution of the occupation total, and expected occupation total, at any particular lattice point. The limiting forms of these results provide explicit expressions for the corresponding quatities in the case of a binomial random walk having a single absorbing point and, where relevent, in the case of the unrestricted binomial random walk.