Let p be a class of finite soluble groups which is closed under epimorphic images and let g be a saturated formation. Then if G is a group of minimal order belonging to p but not to g, F(G), the Fitting subgroup of G, is the unique minimal normal subgroup of G. It is to groups with this property that the following proposition is applicable.

In his paper [1], W. D. Munn determines the irreducible matrix representations of an arbitrary inverse semigroup. Munn also gives a necessary and sufficient condition upon a 0-simple inverse semigroup for it to have a non-trivial matirx representation and for such semigroups gives a complete account of their representations. Munn's results rest upon the earlier work of Clifford [2] in which the representations of Brandt semigroups were determined. An alternative account of such representations was given by Munn in [3]. This earlier work is presented in Sections 5.2 and 5.4 of [4].

The object of this note is to study two properties of groups, which we will denote by (*) and (**). The property (*) is possessed by solvable groups (and in fact, by groups which have a solvable invariant system) and the property (**) is possessed by nilpotent groups (and in fact, by groups which have a central system).

In this paper, necessary and sufficient conditions are established for the continuation of small oscillations of a simple pendulum as its string length is decreased uniformly to zero, and the accompanying terminal value of the angular displacement is determined.

This note shows that the set of bare points of a compact convex subset of a normed linear space is, in general, a proper subset of its set of exposed points.

Throughout this paper F is an algebraically closed field of characteristic p (≠ 0) and g is a finite group whose order is divisible by p. We define in the usual way an F-representation of g (or F G-representation) and its corresponding module. The isomorphism class of the, F G-representation module M is written {M} or, where no confusion arises, M. A (G) denotes the F-representation algebra of G over the complex field G (as defined on pages 73 and 82 of [6]).

The generalized Pexider equation where f and g are unknown and x, y, are real, has been discussed by J. Aczél [1] and J. Aczé and M. Hosszú [2]. In [2] it is shown that if F is continuous and F and H are strictly increasing in their first variables and strictly decreasing in their second variables, then two initial conditions suffice to determine at most one continuous solution f of (1). We extend these results to strictly increasing and strictly decreasing functions F and derive results for strictly monotonic F and H.

A sequence a1 < a2 < … of positive integers is said to be primitive if no element of the sequence divides any other. The study of primitive sequences arose naturally out of investigations into the subject of abundant numbers, where sequences each of whose elements is of the form , the pi being fixed primes, are of particular importance. Such a sequence is said to be built up from the primes p1…pr. Thus Dickson [1], in an early paper on abundant numbers, proved that a primitive sequence built up from a fixed set of primes is necessarily finite.

This paper considers properties of the spectrum of differential operators derived from differential expressions of the second order. The object is to link the spectral properties of these differential operators with the analytic, function-theoretic properties of the solutions of the differential equation. This provides an alternative approach to the spectral theory of these differential operators but one which is consistent with the standard definitions used in Hilbert space theory. In this way the approach may be of interest to applied mathematicians and theoretical physicists.

A right [left] unitary S-system is a set M with right [left] operators in a semigroup S with 1, where x1 = x [1x = x] for all x ∈ M. We define a semigroup S with 1 to be completely right [left] injective provided that every right [left] unitary S-system is injective. The main purpose of this paper is to determine a structure for completely right [left] injective semigroups whose idempotents are in the centre.

In this paper elementary properties are established for matrices whose coordinates are elements of a lattice L. In particular, many of the results of Luce [4] are extended to the case where L is an orthomodular lattice, a lattice with an orthocomplementation denoted by in which a ≦ b ⇒ a ∨(a′ ∧ b) = b. Originally, these were called orthocomplemented weakly modular lattices, Foulis [2]. In Theorem 1 a characterization is given of the nucleus with respect to matrix multiplication, which is in general nonassociative. Matrices with A-1 = transpose (A) are characterized in Lemma 8. Theorems 3 and 4 respectively, give partial characterizations of zero divisors and inverses.

For a series of elements of a topological vector space, necessary and sufficient conditions are found, in terms of the set of finite partial sums, for unconditional convergence and for the corresponding Cauchy condition. The extent to which these results remain valid for topological groups is investigated. A new and direct proof, for locally convex spaces, is given of the theorem of Orlicz.

The purpose of this paper is twofold. In [6] Tomiuk gives a representation theorem for a topologically simple right complemented algebra that is also an annihilator algebra. We strengthen this and then give a converse, so as to characterise right complemented algebras among respectively primitive Banach algebras and primitive annihilator Banach algebras. Our second aim is to investigate the relationship between the different annihilator conditions—left annihilator, right annihilator, annihilator, and dual—when imposed on a complemented algebra. Tomiuk [6] has already shown that a right complemented semisimple algebra that is a left annihilator algebra is an annihilator algebra; further, a topologically simple bi-complemented algebra that is also an annihilator algebra is dual. We show that for a topologically simple right complemented algebra all four annihilator conditions are equivalent. Further, for a semi-simple Banach algebra the first three are equivalent provided it is right complemented, and if it is also left complemented, then they are equivalent to duality.

The general paraxial particle-optic properties of the hexapole field are summarized and lens formulae derived. Formulae are given for the transmission as a function of particle velocity and system geometry for geometries appropriate to the case of neutral atom beams and of slow neutron beams respectively. These formulae are used to evaluate the hexapole magnet as a velocity selector and polarizer for atomic beams and as a spin polarizer for neutron beams. Some experimental observations on potassium beams are quoted in support of the theory.

In [8] and [9] we initiated a study of lattice theory by means of Baer semigroups. Basically, a Baer semigroup is a multiplicative semigroup with 0 in which the left annihilator L(x) of each element x is a principal left ideal generated by an idempotent, while its right annihilator R(x) is a principal right ideal generated by an idempotent. By [8, Lemma 2, p. 86], L(0) has a unique idempotent generator 1 which is effective as a two-sided multiplicative identity for S. For any Baer semigroup S, if we use set inclusion to partially order both ℒ = ℒ (S) = {L(x) | x ∈ S} and ℛ = ℛ (S) = {R(x) | x ∈ S}, we have by [8, Theorem 5, p. 86], that ℒ and ℛ form dual isomorphic lattices with 0 and 1. The Baer semigroup S is said to coordinatize the lattice L in case ℒ(S) is isomorphic to L. In connection with this, it is important to note that by [9, Theorem 2.3, p. 1214], a poset P with 0 and 1 is a lattice if and only if it can be coordinatized by a Baer semigroup.

In [1], the natural representation module of the symmetric groups, hereafter called the first natural representation module of the symmetric groups, was analysed. It is the purpose of this paper to analyse the second natural representation module of the symmetric groups.

In the literature concerning matrices whose co-ordinates are elements of a Boolean lattice, one may find three different definitions for the determinant of a matrix. We shall call these the first, second and third determinant and will denote the value of the ith determinant of a matrix A by |A |i for i = 1, 2, 3. The first determinant may be defined for square matrices over an arbitrary lattice. The second and third determinants may be defined for square matrices over any lattice L with a greatest element I, a least element o and an orthocomplementation′: L→L, that is a′ is a complement of a, a = a″ and a ≤ b implies that b′ ≤ a′ for all a, b in L. In this paper we obtain some elementary properties of these determinants in this general setting and in the particular case where L is an orthomodular lattice, that is a lattice with o, 1 and an orthocomplementation' such that

A second method for the synthesis of 8-bromo-3-methoxyfluoranthene from 2-bromo-7-methoxyfluorene is described. The autoxidation of fluorene derivatives including the 9-carboxylic acids and of related compounds, particularly 4-α-cyanobenzyl-7−oxo-7H-benz[de]anthracene, is discussed. Unsuccessful attempts to prepare 3-bromofluorene-9-acetic acid in good yield are reported.