The purpose of this note is to improve Theorem 1·1 of (1) to the following result:

Theorem 1. Let H be the ordinary free product of a sequence of groups Gn, and let m be any integer > 0. If

where each gi є Gi and is not the identity, then h is not equal to any product of fewer than m commutators from H.

In the classical theory of representations of a finite group by matrices over a field , the concept of the group algebra (group ring) over is of fundamental importance. The chief property of such an algebra is that it is semi-simple, provided that the characteristic of is zero or a prime not dividing the order of the group. As a consequence of this, the representations of the algebra, and hence of the group, are completely reducible.

Introduction. Dans cet article, K désigne un corps commutatif quelconque. Toutes les algèbres de Lie considérées sont des algèbres de Lie de dimension finie sur K. Si g est une algèbre de Lie, on désigne par (D0g, D1g, D2g, …) la suite des algèbres dérivées de g (D0g = g; Dig = [Di−1g, Di−1g]). Si h est un idéal de g, et si x ↦ g, on désigne par adh, x l'application y → [x, y] de h dans h.

Let L be a Lie ring and denote the product of x and y in L by [x, y]. The ring L is said to satisfy the Engel condition (cf. (1)), if for every pair of elements x, yεL there is an integer k = k(x, y)such that

If k(x, y) can be taken equal to a fixed integer n for all x, y ε L then L is said to satisfy the n-th Engel condition.

The purpose of the present note is to prove the following two theorems:

Theorem 1. Let Q be an equicharacteristic local domain with maximal ideal m. Let a be any ideal of Q. Then the intersection of all integrally closed m-primary ideals of Q which contain a is the integral closure ā of a.

Theorem 2. If Q is as above, and if S denotes the set of valuations on the field of fractions F of Q which are associated with Q, then the intersection of the valuation rings belonging to valuations in S is the integral closure of Q.

This paper describes a generalization of the inverse of a non-singular matrix, as the unique solution of a certain set of equations. This generalized inverse exists for any (possibly rectangular) matrix whatsoever with complex elements. It is used here for solving linear matrix equations, and among other applications for finding an expression for the principal idempotent elements of a matrix. Also a new type of spectral decomposition is given.

In most theories for the construction of finite groups with given properties a major difficulty is the ‘isomorphism problem’, which consists of specifying how one representative of each class of isomorphic groups may be selected from the totality of groups constructed by the process laid down. To do this we need a practical criterion for the isomorphism of two constructed groups. The main object of the present paper is to establish such a criterion in a particular case, which in spite of its simplicity is important because it gives a method for the construction of A-groups (i.e. soluble groups whose Sylow subgroups are all Abelian). All soluble groups of cube-free order are included in this class of groups, and to exemplify the application of the criterion we summarize that part of our unpublished dissertation (7) which deals in detail with the groups of order 22.32.52.

The so-called AF + BΦ theorem of Max Noether refers to a whole collection of results. In most of these, the object is to determine conditions under which a form H(x, y, z) will belong to the ideal generated by two given forms F(x, y, z) and Φ(x, y, z). This problem is of obvious importance in the theory of plane curves, and therefore it is customary to state the conditions in geometric language. To give one example, a particularly important form of the theorem may be stated as follows.*

It is well known that a characteristically-simple finite group, that is, a group having no characteristic subgroup other than itself and the identity subgroup, must be either simple or the direct product of a number of isomorphic simple groups. It was suggested to the author by Prof. Hall that finite groups possessing exactly one proper characteristic subgroup would repay attention. We shall call a finite group having a unique proper characteristic subgroup a ‘UCS group’. In the present paper we first give some results on direct products of isomorphic UCS groups, and then we consider in more detail one of the types of UCS groups which can exist, that consisting of groups whose orders are divisible by exactly two distinct primes.

1. Let H = (aij) be a positive-definite Hermitian matrix of order n. For any k distinct integers i1, i2, …, ik between 1 and n, we shall use the symbol (i1, i2, …, ik) to denote the k-rowed principal submatrix of H corresponding to the rows and columns with indices i1, i2, …, ik. It is well known thatM

and more generally,

1. This note takes its origin in a remark by Brauer (1) and Perfect (5): Let A be a square complex matrix of order n whose characteristic roots are α1,…, αn. If X1 is a characteristic column vector with associated root α and k is any row vector, then the characteristic roots of A + X1 k are α1 + KX1, α2, …, αn. Recently, Goddard (2) extended this result as follows: If x1; …, xr are linearly independent characteristic column vectors associated with the characteristic roots α1, …, αr of the matrix A, whose elements lie in any algebraically closed field, then any characteristic root of Λ + KX is also a characteristic root of A + XK, where K is an arbitrary r × n matrix, X = (x1, …, xr) and Λ = diag (α1, …, αr). We shall prove some theorems of which these and other well-known results are special cases.

In the first part of this paper we consider partially ordered sets for which the intrinsic ‘interval topology’ ((1), p. 60) is Hausdorffian. The main result of this section is the following: the interval topology of a lattice is Hausdorffian if and only if convergence with respect to the interval topology is equivalent to strong o*-convergence. This may be regarded as an answer either to Birkhoff's problem 23 ((1), p. 62) or to his problem 25 ((1), p. 64). In the case of a complete lattice, we have an alternative formulation. The interval topology of a complete lattice is Hausdorffian if and only if every filter (or, alternatively, every directed net) in the lattice has an o-convergent refinement. This condition may be regarded as a strong type of compactness.

In this note I obtain bounds for the least integral solutions of the equation

in terms of

For ternary diagonal forms, such bounds have been given by Axel Thue (4) and, more recently, by Holzer (1), Mordell (2) and Skolem (3), but these lead only to bad estimates for general ternaries. So far as I know there have not been given estimates for n≽4. Here I generalize Thue's method to prove:

Theorem. Suppose that n ≥ 2 and that f(ɛ) represents zero. Then there is an integral solution of f(a) = 0 with

In general, the class of a nilpotent linear algebra of dimension n is at most n + 1, and the index, or derived length, of a solvable linear algebra of dimension n is at most n. In this note it is shown that, for a nilpotent linear algebra of dimension n satisfying x2 = 0 for all x, the class is at most n; and bounds are obtained for the indices of solvable Lie algebras.

Let Z denote the extended complex plane with the natural topology. That is, Z consists of the finite complex plane Zf together with a point, denoted by ∞, any neighbourhood of which consists of the union of this point and the complement of a bounded set of ZF. Let Z* = [z │ z ↦ Z, z ╪ 0, z ╪ ∞]. Thus Z is homoeomorphic to a 2-sphere and Z* is homoeomorphic to a cylinder.

A number of methods are at present available for the numerical solution of linear ordinary differential equations over a restricted range of the independent variable. Of these perhaps the most commonly used are methods based on finite differences which have as their aim the construction of a table of values of the dependent variable at stated intervals of the independent variable. The appropriate method to be used depends upon the boundary conditions. Thus for second-order equations, for example, if boundary conditions occur at each end of the range of integration the relaxation method of Southwell (6) is suitable, whereas if this is not the case resort may be made to step-by-step methods or the recently proposed method of Allen and Severn(1). This method, applicable to equations of any order, brings the problem within the scope of normal relaxation methods at the expense of an increase in the order. In all such methods the accuracy attainable for a given (small) amount of labour is a worth-while consideration, and methods have been suggested by Fox (3) and Fox and Goodwin (4) for improving this by the use of comparatively large intervals of the independent variable.

In a recent joint paper (1) with Prof. Besicovitch we announced the conjecture that for almost all one-dimensional Brownian paths, the set of zeros has dimensional number ½, and zero A½-measure. It is the purpose of this paper to give a proof of this result. In doing so we consider the graph C(ω) of a Brownian path ω as a point set in the plane, and prove that, with probability 1, C(ω) has dimensional number ¾ and zero Λ¾-measure.

The Minkowski–Hlawka theorem† asserts that, if S is any n-dimensional star body, with the origin o as centre, and with volume less than 2ζ(n), then there is a lattice of determinant 1 which has no point other than o in S. One of the methods used to prove this theorem splits up into three stages, (a) A function ρ(x) is considered, and it is shown that some suitably defined mean value of the sum

taken over a suitable set of lattices Λ of determinant 1, is equal, or approximately equal, to the integral

over the whole space. (b) By taking ρ(x) to be equal, or approximately equal, to

where σ(x) is the characteristic function of S, and μ(r) is the Möbius function, it is shown that a corresponding mean value of the sum

where Λ* is the set of primitive points of the lattice Λ, is equal, or approximately equal, to

The method of solving equations by iteration is very old and is discussed in many well-known books. But conditions for its validity have never been properly formulated. In the first place, it is necessary to know that the method will not carry us outside the domain of definition of our functions, and that the ‘approximations’ will not converge to something which is not a solution of the equations. These difficulties are easily forestalled; it is more difficult to ensure that the process really will converge. Our object will be to find the most general conditions under which we can set off with the certainty that we will ultimately arrive at a root, in the case of one equation with one unknown.

Certain stochastic processes with discrete states in continuous time can be converted into Markov processes by the well-known method of including supplementary variables. It is shown that the resulting integro-differential equations simplify considerably when some distributions associated with the process have rational Laplace transforms. The results justify the formal use of complex transition probabilities. Conditions under which it is likely to be possible to obtain a solution for arbitrary distributions are examined, and the results are related briefly to other methods of investigating these processes.