In this paper we obtain various sufficient conditions for the solubility of a finite group. In particular, we show that if G is a finite group and p≥5 is a prime such that all p′-subgroups of G are nilpotent, then G is soluble. We show also that if G is a finite group which has a cyclic Sylow p-subgroup Pand such that for all p′-subgroups H of G, H is nilpotent and H′ is cyclic, then, if p≠3, either P◃G or G has a normal p-complement.

The concept of a local system of a set W is defined in ((8), p. 166) and ((12), p. 126). Recall that a set ℒ of subsets of W is a local system if Uℒ = W and ℒ is directed in the following sense: for every finite set H1, …, Hn of elements of ℒ, there is an M∈ℒ such that Hi⊂M for 1≤i≤n. If Σ is a class of groups, L(Σ) is the class of all groups G that possess a local system of Σ subgroups. Σ satisfies the local theorem, or is L-closed, if L(Σ)⊂Σ. Many classes of groups which satisfy the local theorem are discussed in ((12), pp. 126–144).

In (1) A. H. Boers has studied the properties of n-PA rings, i.e. rings where any n-product (any product of n factors) is independent of its bracketing; a private communication is quoted in that paper to the effect that the n-PA rings were defined by Y. Ilamed starting from problems in mathematical physics.

Posner (1) proved the following result:

Theorem. Let R be a prime ring and d a derivation of R such that, for all a∈R, ad(a)–d(a) a∈Z (centre of R). Then, if d is not a zero derivation, R is commutative.

The purpose of this note is to give a short proof of this theorem. We state the following known result ((1), Lemma 1).

Let k be a field of characteristic other than 2 and let g(k) denote the multiplicative group k* of the field k modulo squares, i.e. g(k) = k*/k*2. This is an abelian group of exponent 2 and its order, if finite, is a power of 2. We denote by G(k) the Grothendieck group of quadratic forms over k.

Clifford's well-known chain of theorems concerning circles in a plane may be stated as follows (the first two links being trivial):

(C2) Given any point A, and through it two circles C1 and C2, these meet again in a point A12, say.

(C3) Given three circles C1, C2, C3 through A, the three points A23, A13, An lie on a circle C123, say.

(C4) Given four circles C1,…,C4 through A, the four circles C234,…,C123 meet in a point A1234 (Wallace(1)).

(C5) Given five circles C1,…,C5 through A, the five points A2345,…,A1234 lie on a circle C12345 (Miquel (2)).

And so on (Clifford(3)).

Grace (4) showed that in three dimensions [3], there is no direct analogue to Clifford's chain, but in [4] he noted the following:

(G4) Given any point A in [4], and through it four hyperspheres S1,…, S4, these meet again in a point A1234, say.

(G5) Given five hyperspheres Sl,…,S5 through a point A, the five points A2345, A1234 lie on a hypersphere S12345, say.

(G6) Given six hyperspheres Sl,…, S7 through A, the six hyperspheres S23456,…, A12345 have in common a point A123456.

This leads naturally to the conjecture

(G7) Given seven hyperspheres Sl,…, S7 through A, the seven points A234567,…, A123456 lie on a hypersphere S1234567.

Let ΦSr(X) be the stable homotopy group

where SnX means the n-fold suspension of X. For example, the groups ΦSr(S0) are the stable homotopy groups of spheres. Let

be the ‘infinite-dimensional’ orthogonal group. Then topologists are familiar with the ‘stable J-homomorphism’

G. W. Whitehead observed that J factors through an ‘even more stable’ J-homomorphism

he conjectured that J′ is epi (for r > 0).

1. Let G be a compact Abelian group with character group X. Let be an increasing sequence of finite symmetric subsets of X, and consider a symmetric subset P of . For any Hermitian complex-valued function u on P, we write snu for the real-valued trigonometric polynomial . Edwards, Hewitt and Ross(4) investigated the following property for a non-void measurable subset W of G satisfying W ⊂ (int W)−:

The validity of this implication was shown to be independent of the choice of . Accordingly, if (*) holds, P is called an FZ(W)-set. If P is an FZ(W)-set for all W, then P is termed a full FZ-set or full Fatou-Zygmund set. In this paper, we characterize the full FZ-sets as FZ(G)-sets satisfying a certain algebraic condition. In particular, we show that if G is connected, then a symmetric subset of X is an FZ(G)-set if and only if it is a full FZ-set. Some of the techniques are adaptations of those of Mme Déchamps-Gondim(1), (2). The class of full FZ-sets is not always closed under the operation of finite unions; this contrasts with the situation for Sidon sets and for FZ(G)-sets.

The union of 2 disjoint sets of interpolation need not be of interpolation even if every finite subset is of interpolation with bounded constant.

1. Introduction. In a recent paper (2) Daykin has made the Conjecture. If t is real, then

where

He has proved (1, 2) that (i) Φ(t) is a convex function of t, (ii) Φ(t)≥ n if t ≥ 0 or 2, and (iii) Φ(t) takes values arbitrarily close to ½n if n is even, x2 = x4 = … = xn = 1, and xl = x3 = … = xn-1 and t have suitable small positive values. It is known (3) that (1) is true if t = 1.

If the series Σαn is convergent, it follows that Σαnzn is uniformly convergent in a Stolz angle at z = 1 ((7), p. 229). It was shown in (3) that for the Ces methods (C, α) with α > – 1, and for a certain class of general Nörlund means (containing (C, α) for 0 < α ≤ 1), summability of Σαn implies uniform summability in a Stolz angle at z = 1. In section 2 we prove this theorem for a class of methods which satisfy a mean value property ((6), p. 31), and for a wider class of Nörlund methods which includes (C, α) for all α > 0. An analog is also proved for absolute summability.

In (6) Plancherel considered linear, bounded transforms which satisfy

where the linear, bounded transform is defined by

Among other things he proved: if W1, W2 and W3 are linear, bounded transforms satisfying (1), then

One of the topics covered by most textbooks on partial differential equations is the classification problem for second-order equations in the plane. In a typical treatment, it is shown that under some smoothness conditions on the coefficients, hyperbolic and parabolic equations can be reduced locally to normal form (uniformized) by an elementary procedure. For elliptic equations, the procedure fails unless an extraneous hypothesis (analyticity of the coefficients) is introduced. It is then pointed out that a different and much deeper method (essentially the general uniformization theorem) is effective for the elliptic case and even yields a global result. It is striking, but not surprising in view of recent developments on generalized solutions, that the alternate (global) procedure for elliptic equations requires much less smoothness of the coefficients than is needed for a sensible local result in the other cases.

A method of finding explicit expressions for the roots of a certain class of transcendental equations is discussed. In particular it is shown by determining a canonical solution of an associated Riemann boundary-value problem that expressions for the roots may be derived in closed form. The explicit solutions to two transcendental equations, tan β = ωβ and β tan β = ω, are discussed in detail, and additional specific results are given.

We shall consider a non-stationary Markov chain on a countable state space E. The transition probabilities {P(s, t), 0 ≤ s ≤ t <t0 ≤ ∞} are assumed to be continuous in (s, t) uniformly in the state i ε E.

By the use of a semi-martingale property of the Kolmogorov supremum, the results of Pyke (6) on the weak convergence of the empirical process with random sample size are simplified and extended to the case of p(≥1)-dimensional stochastic vectors.

Positive definite functions on metric spaces were considered by Schoenberg (26). We write σk for the unit hypersphere in (k + 1)-space; then σk is a metric space under geodesic distance. The functions which are positive definite (p.d.) on σk were characterized by Schoenberg (27), who also obtained a necessary condition for a function to be p.d. on the it sphere σ∞ in Hilbert space. We extend this result by showing that Schoenberg's necessary condition for a function to be p.d. on σ∞ is also sufficient.

An approximation is found to the solution of the partial differential equation

in the region −1 ≤ x ≤ 1, t > 0, where u satisfies a general linear boundary condition on x = ± 1. This approximation is a polynomial in x, and is an exact solution of a perturbed form of the differential equation. By choosing the perturbation appropriately, this approach is mathematically equivalent to some recent methods for solving the differential equation in the form of a Chebyshev series. Better approximations to the required solution (and particularly to the eigenvalues) are obtained by choosing the perturbation to satisfy a least squares criterion.

The variational derivative of Lagrange densities which are functions of the metric tensor and its first and second derivatives is considered. This tensor is generally of fourth order in the derivatives of the metric tensor. If derivatives of any order are not present in the variational derivative then the Lagrange density is said to be degenerate in these derivatives. An explicit expression for the variational derivative of Lagrange densities which are degenerate in third and/or fourth derivatives is displayed in tensor form.

1. An isotropic tensor is one the values of whose components are unaltered by any rotation of rectangular axes (with metric σi(dxi)2). Those up to order 4 in 2 and 3 dimensions have many applications. The results suggest a general theorem for tensors of order m in n dimensions, that any isotropic tensor can be expressed as a linear combination of products of δ and є tensors, where δij = 1 if i = j and 0 otherwise, and is 0 if any two of the i1 to in are equal, 1 if i1…in is an even permutation of 1, 2, 3, …,n, and – 1 if it is an odd permutation.