In this paper we study the structure of the α-central series of the nilpotent wreath product of two Abelian groups, the α-central series being the intersection of the upper central series with the base group. Let C = A wr B, the standard restricted wreath product of A and B. Then Baumslag(1) showed that C is nilpotent if and only if A is a nilpotent p-group of finite exponent and B is a finite p-group for the same prime p. In (5) Liebeck calculated the nilpotency class of the nilpotent wreath product of two Abelian groups. We obtain an expression for an element of the base group to belong to a given term of the upper central series.

This paper examines the relationship between extensions in a variety and general extensions in the category of associative algebras. Our associative algebras are all unitary, over some fixed commutative ring Λ with identity, but while our discussion will be restricted to this category, it is clear that obvious analogues exist for groups, Lie algebras and Jordan algebras. (We use the notion of a bimultiplication of an associative algebra. In (2), Knopfmacher gives the definition of a bimultiplication in any variety of linear algebras.)

In this note (cf. sections 3, 4) I shall give an axiomatization of those fields (of characteristic ≠ 2) which have a theory of quadratic forms like the -adic numbers or like the real numbers. This leads then, for instance, to a generalization of the well-known theorems on -adic forms to a wider class of fields, including non-local ones. The main purpose of the exercise is, however, to separate out the roles of the arithmetic in the underlying field, on the one hand, which solely enters into the verification of the axioms, and of the ordinary algebra of quadratic forms on the other hand. The resulting clarification of the structure of the theory is of interest even in the known -adic case.

In this note we prove the following result due to Milnor ((5), Theorem 4·1).

This theorem proves the equivalence of two conjectures on perfect Singer difference sets, one of which is concerned with the actual computation of the set, and the other being of interest in the theory of multipliers and fixed sets.

Multipliers of a difference set are of great importance in existence theorems, since they enable us to reject many configurations en bloc. (For a description of such theorems, see Mann (1).) The following theorem, which determines those cyclic group difference sets for which −1 is a multiplier, has been proved before by different methods (see, for example, Yamamoto(2) and Johnsen(3); and a more elementary matrix proof by Brualdi(4)) but the following ‘elementary’ proof may be of interest.

Recently Isbell(4) and Ward (6) have given examples of pairs of uniformities inducing the same topology on the hyperspace of subsets of a uniform space. Smith (5) has shown that, if two uniformities induce the same topology on the hyperspace, they must be equal in proximity. The author, in (2), has introduced a relation, ‘height’, between uniformities which is in a sense dual to proximity. It is the purpose of this paper to show that, if two uniformities are equal in proximity and height, then they induce the same topology on the hyperspace. Although Ward's uniformities are not equal in height, an example of two distinct uniformities equal in proximity and height is given in (3). In fact, a slight modification of that example will give an example of a countably infinite number of distinct uniformities, all of which are equal in proximity and height.

Let X be a compact differentiable manifold of dimension 2m. A differentiable map from X to euclidean (2m + t)-space is an immersion if its Jacobian has rank 2m at each point of X; it is an embedding if it is also one–one. The existence of such an embedding or immersion implies that the characteristic classes of X satisfy certain integrality conditions; these can be used to obtain lower bounds for the integer t. In a similar way many other geometric properties of X can be deduced from a single integrality theorem involving characteristic classes of various vector bundles over X (see for instance (5)).

L-structure in a Banach space X was defined in (3) by L-projections, that is projections P satisfying

for all x ∈ X. The significance of L-structure is shown by the following facts: (1) All L-projections on X commute and together form a complete Boolean algebra. (2) X can be isometrically represented as a vector-valued L1 on a measure space constructed from the Boolean algebra of its L-projections (2). (3) L1-spaces in the ordinary sense are characterized among Banach spaces by properties equivalent to having so many L-projections that the representation in (2) is everywhere one-dimensional.

There comes a time in the development of a cohomology theory when a discussion of cohomology operations becomes necessary. In the case of complex K-theory, the subject of the present paper, such operations have of course already been investigated by Adams (see (2)), so that any further discussion might appear superfluous. Powerful as Adams's results are, however, the situation still leaves something to be desired: it is not known just what other operations can be defined in K-theory, and it is an inconvenience from the standpoint of stable homotopy theory that Adams's operations are not themselves stable.

D. R. Hughes stated the following conjecture: If π is a finite projective plane satisfying the condition: (C)π contains a collineation group δ inducing a doubly transitive permutation group δ* on the points of a line g, fixed under δ, then the corresponding affine plane πg is a translation plane.

The purpose of this paper and the next is to demonstrate that the ‘perfect Riesz spaces’ of (1) are an effective abstraction of the ‘espaces de Köthe’ of (2). I shall follow the ideas of (1), with certain changes in notation:

If L is a Riesz space and x, y ∈ L, let us denote sup (x, y) by x ∧ y and inf (x, y) by x ∧ y. I shall use the convenient if informal notation xr↓ ((1), section 16·1) and shall in this usage assume that 0 ∈ {r} and that x0 ≥ xτ for all τ. A set A ⊆ L is solid if x ∈ A and |y| ≤ |x| together imply that y ∈ A; A is then balanced. The solid hull of A is the set {y: ∃ x ∈ A, |y| ≤ |x|}; this is the smallest solid set containing A. An ‘ideal’ ((1), section 17) is then a solid subspace.

In this Note I shall give a characterization of ideals of homological dimension one in a local domain, and deduce a result on ideals in a regular local ring of dimension two.

In (2) Janko gives 7 × 7 matrices over GF (11) which generate a copy of his new simple group. This matrix group induces a representation of Janko's group as a collineation group in PG (6,11)—i.e. 6-dimensional protective space over GF (11)—and the present paper is devoted to the results of study of this geometrical representation.

Two well-known numbers associated with a graph G (finite and undirected with no loops or multiple lines) are the (point) chromatic and the line chromatic number of G (see (2)). With G there is associated a graph L(G), called the line-graph of G, such that the line chromatic number of G is the same as the chromatic number of L(G). This concept was originated by Whitney (9) in 1932. In 1963, Sedlâček (8) characterized graphs with planar line-graphs. In this note we introduce the notions of the total chromatic number and the total graph of a graph, and characterize graphs with planar total graphs.

It is shown that for k ≥ 5 in every k-chromatic graph there is a set of k distinct vertices V1, …, Vk with the property that for i, j = 1, …, k and i ≠ j the graph contains the union of a set of 4 paths connecting Vi and Vj no two of which have any edge or any vertex besides Vi and Vj in common and k − 1 paths connecting Vi and Vj no two of which have any edge in common.

It is generally conjectured that if α1, α2 …, αk are algebraic numbers for which no equation of the form

is satisfied with rational ri not all zero, and if K > 1 + l/k, then there are only finitely many sets of integers p1, p2, …, pkq, q > 0, such that

This result would be best possible, for it is well known that (1) has infinitely many solutions when K = 1 + 1/k. † If α1, α2, …, αk are elements of an algebraic number field of degree k + 1 the result can be deduced easily (see Perron (11)). The famous theorem of Roth (13) asserts the truth of the conjecture in the case k = 1 and this implies that for any positive integer k, (1) certainly has only finitely many solutions if K > 2. Nothing further in this direction however has hitherto been proved.‡

Let f be a 2π-periodic function of the class L(−π,π). Put

We call, with Žuk(6), the quantity L(p)(h, f) the L-modulus of smoothness of order p of the function f. Žuk has recently obtained, in (5) and (6), generalizations of a number of classical results on the absolute convergence of Fourier series, as also on the order of Fourier coefficients by employing the concept of the L-modulus of smoothness which is obviously a more general concept than that of the modulus of continuity. It is the purpose of this note to prove a theorem on the almost everywhere convergence of Fourier series of f involving the concept of L(p)(h, f).

1. Given an infinite series ∑an, we consider its Nörlund transformation

where

It is know that if

then the (N, p)-mean reduces to the familiar (C, α)-mean.

1. The consequences of assuming that an integral function or its derivatives take integer values for some or all integer values of the variable have been widely investigated. The best-known result of this kind was obtained by Pólya(9,10) and Hardy(6) in 1920 and states that among all transcendental integral functions which assume integer values for all non-negative integer values of the variable that of least increase is the function 2Z (cf. also (7)). Analogous results relating to integers in the Gaussian field were proved by Fukasawa(3)in 1926 and Gelfond(4)in 1929, and theorems relating to the function and its derivatives are included in the works of Gelfond (5), Selberg(14) and Straus (15). (For further results and references see (1,2, 11).) The earlier of these investigations may be regarded as the genesis of the celebrated Gelfond-Schneidersolution to the seventh problem of Hilbert and the much important work arising therefrom.