We call two measures equivalent if each is absolutely continuous with respect to the other (cf. [1]). Let G be a locally compact topological group and let μ be a non-negative Baire measure on G (i.e. μ is denned on all Baire sets, finite on compact sets and positive on open sets). We say that μ is stable if μ (E)=0 implies μ(tE)=0 for each t ∈ G. A. M. Macbeath made the conjecture that every stable non-trivial Baire measure is equivalent to the Haar measure. In this paper we prove the following slightly stronger result:

Theorem. Every stable non-trivial measure defined on Baire sets and finite on some open set is equivalent to the Haar measure.

The existence of a function g of hhaving the property that pr divides the order of the automorphism group of a finite group G whenever pg divides the order of G was first established by Ledermann and Neumann [4], who showed that the least such function g(h) satisfies the inequality

Later Green [2] improved this estimate to

In the Present paper this will be revised, for sufficiently large h, to

Let G be a given group and A, B be two subgroups of G which may or may not coincide. A homomorphism μ which maps A onto B is called a partial endomorphism of G. When A coincides with G then we call μ a total endomorphism or as it is usually called an endomorphism of G. If μ* is a partial (or total) endomorphism of a supergroup G* ⊇ G, then we say that μ* extends, or continues, μ when μ* is defined for at least all the elements a ∈ A and moreover aμ = aμ* for all a ∈ A If the partial endomorphism μ is an isomorphic mapping then we speak of a partial automorphism of G.

In this paper we evaluate some integrals involving U-functions by the methods of the Operational Calculus. The results obtained are quite general and many of them include, as particular cases, some known results.

A function ψ (p) is operationally related with another function f(t), if they satisfy the integral equation

2. Theorem. If

and

As usual, we shall denote (1) by the symbolic expression

provided that the integral is convergent. HereR(α) > 0, R(p) > 0, n = 2,3,4, …, andmeans that in the expression following it, i is to bee replaced by – i and the two expressions are to be added.

In this paper three integrals involving E-functions are evaluated in terms of E-functions. The formulae to be established are:

where n is a positive integer, | args z < π, R(γ ± m ÷ ½) > 0, αρ+ν = (2γ + ν)/2n (ν = 1, 2, …, 2n), αρ+2n+i = (γ + m - ½ + i)/n, αρ+3n+i = (γ - m - ½ + i)/n, βα+i = (γ + κ + i)/n, βα+ν+i = (γ - κ + i)/n(i = 1, 2, …, n).

where n is a positive integer, |arg z| < π, R(λ±μ±ν) > 0, αp+i+1 = (λ + μ + ν + i)/n, αp+n+i+1 = (λ - μ + ν + i)/n, αp+2n+i+1 = (λ + μ - ν + i)/n, αp+3n+i+1 = (λ - μ - ν + i)/n (i = 0, 1, 2, …, n - 1), βa+i+1 = (2λ + j)/2n (j = 0, 1, 2, …, 2n - 1).

where n is a positive integer, R(λ) > ½, |arg z| < π, αp+i+1 = (2λ - 1 + i)/2n (i = 0, 1, 2, …, 2n-1), βq+j+1 = (λ + μ j)/n, βq+n+j+1 = (λ - μ + j)/n (j = 0, 1, 2, …, n-1).

Definitions and introduction. Let Ц = {Ui|i ∈ I} be a system of subsets of a normal topological space R; i.e. a mapping from the index set I into the set of all subsets of R. The order of a point x is the number of distinct member sets of Ц which contain x, and is denoted by x: Ц; the sets Ui are here considered distinct if they have distinct indices. Thus x: Ц is the number of indices i for which x ∈ Ui; ν(Ц) = max {x: Ц | x ∈ R} is called the order of the system Ц. If every point has an (open) neighbourhood meeting only finitely many members of Ц, then Ц is said to be locally finite.

This note discusses the determination of the coefficients an in the dual trigonometrical series

where p = ± 1 and F(x), G(x) are prescribed functions of X. It is shown that this problem and the corresponding one in which the sines in equations (1) are replaced by cosines are easily reduced to a form in which the results I have recently given in this journal [1] may be applied.

As with my previous paper on this subject, the analysis is purely formal and no attempt is made to give precise conditions for which the solution is valid.

Potential problems in which different conditions hold over two different parts of the same boundary can often be conveniently reduced to the solution of a pair of dual integral equations. In some problems, however, the boundary condition is such that different conditions hold over three different parts of the boundary and, in such cases, the integral equations involved are frequently of the form

where f(r), g(r) are specified functions of r, p = ± ½ and ø(u) is to be found. Such equations might well be called triple integral equations and, in this note, I point out certain special cases which I have found to be capable of solution in closed form.

This note extends the concept of the inner automorphism, but here applies only to those finite groups G for which some member of the lower central series is Abelian. In general (e.g. when G is metabelian) the construction yields an endomorphism semigroup, but in the special case where Gis nilpotent (and may therefore, for our present purposes, be considered as a p-group) a group of automorphisms results.

Let G be a locally compact topological group, let H be a closed subgroup and let G/H be the space of left cosets = xH with the natural topology. We denote by μ a non-negative measure in G/Hdefined on the ring of Baire sets. G acts by left multiplication as a transitive group of homeomorphisms on G/H: Every t ∈ G defines the homeomorphism We write, for E ⊂ G/H, tE = . The measure μ is called stable (cf. [3], [4]) if from t ∈ G, E ⊂ G/H and μ(E) = 0 follows μ(tE) = 0. We say that μ is locally finite [3], [5] if every set of positive measure contains a subset of positive finite measure.