This paper is devoted to the construction of faithful representations of the automorphism group and the holomorph of an extension of an abelian group by some other group, the representations here being homomorphisms into certain restricted parts of the automorphism groups of smallish abelian groups. We apply these to two very specific cases, namely to finitely generated metabelian groups and to certain soluble groups of finite rank. We describe the applications first.

Let R be a commutative ring which may not contain a multiplicative identity. A set of elements a1,…,ak in R will be called an H1-set (this notation is explained in section 1) if for each relation r1a1 + … +rkak = 0 (ri ∈ R) there exist elements sij ∈ R such that

where Xl,…,Xk are indeterminates. Any R-sequence is an H1-set, but there do exist H1-sets which are not R-sequences (see section 1). Throughout this note we consider an H1-set a1,…,ak which we suppose to be partitioned into two non-empty sets bl…, br and cl,…, cs. Our main purpose is to show that the ideals B = Rb1 + … + Rbr and C = Rc1 + … + Rcs satisfy Bm ∩ Cn = BmCn for all positive integers m and n (Corollary 1). This generalizes Lemma 2 of Caruth(2) where the result is proved when a1,…, ak is a permutable R-sequence. Our proof involves more detail than is necessary just for this, and we obtain various other properties of H1-sets. In particular we extend the main results of Corsini(3) concerning the symmetric and Rees algebras of a power of the ideal Ra1 +… + Rak (Corollary 3).

This paper is a continuation of (F3). In its first part we shall expand and extend the general theory of the earlier paper, while in the second part we specialize to number fields. The theory of resolvents and of the trace form, presented here, complements the more arithmetic theory of module conductors and module resolvents as described elsewhere (cf. (F4)). Both these papers will be applied in work on the connexion, for tame extensions, between Galois module structure of algebraic integers on the one hand, and Artin conductors and root numbers on the other hand (cf. (F5)). The results of the present paper are however not restricted to the tame case and, it is hoped, will subsequently be applied in a more general context.

An (n, q) graph consists of n labelled nodes and q edges, i.e. unordered pairs of different nodes. The maximum possible value of q is N = ½n(n − 1) and the (n, N) graph is the complete graph. We write B(h, k) = h!/{k!(h − k)!}. The number of (n, q) graphs is clearly B(N, q). A k-cycle in a graph consists of a sequence

of k edges, where D1D2,…, Dk are all different nodes. We are concerned here with the probability P = P(n, q, k) that a k-cycle will be present in an (n, q) graph, that is, the proportion of all (n, q) graphs which contain at least one k-cycle. We investigate the limit of P as n → ∞ and in particular try to determine when P → 1 (when we say that almost all (n, q) graphs contain a k-cycle) and when P → 0 (when we say that almost no (n, q) graphs contain a k-cycle).

The theory of dualizing complexes of Grothendieck and Hartshorne ((5), chapter v) has turned out to be a useful tool even in commutative algebra. For instance, Peskine and Szpiro used dualizing complexes in their (partial) solution of Bass's conjecture concerning finitely-generated (f.-g.) modules of finite injective dimension over a Noetherian local ring ((7), chapitre I, §5); and the present author first obtained the results in (9) by using dualizing complexes.

P. M. Cohn has given a characterization of pseudovaluation on a field in (1). There he defines the notion of an integral element over a multiplicatively closed set and states that the association of the set A to the set A* of all elements that are integral over it is a closure operator. There is an error in the statement that A* = (A*)*. In this note we show, by an example, that A* and (A*)* are not equal in general. We also give an example to show that even for the special type of sets that Cohn needs, namely the gauge sets, this property fails to hold. However, since the other conditions for a closure operator (A ⊆ A* and A ⊆ B implies A* ⊆ B*) are satisfied, one obtains a Kuratowski closure operator by iterating *. If we denote by Ā the closure of A by iterating * then theorem 10·3, its corollary and theorem 10·4 in (1) that have a bearing on this error, are valid as stated with the same proof replacing only K* by . Further theorem 13·3 is valid without modification since Cohn considers only a special class of pseudovaluations namely regular pseudo valuations for which K* = .

It is well-known ((2), Theorem 9·11) that any closed subgroup of Rn is isomorphic (topologically and algebraically) to Ra × Zb, where a, b are suitable non-negative integers. For an infinite product of copies of R, it is also known that any locally compact (hence closed) subgroup is a product of copies R and Z, and that any connected subgroup is a product of copies of R (see (7), (3), respectively). Some information is also given in (3) on closed subgroups of products of copies of R and T, where T = R/Z is the circle group.

It is a well known theorem of Gaschütz (4) and Schenkman (12) that if G is a finite group whose nilpotent residual A is Abelian, then G splits over A and the complements to A in G are conjugate. Following Robinson (10) we describe this situation by saying that G splits conjugately over A. A number of generalizations of this result have since been obtained, some of them being in the context of the formation theory of finite or locally finite groups (see, for example, (1), (3)) and others, for example, the recent and far-reaching results of Robinson (10, 11) being concerned with groups which are not necessarily periodic. Our results here are of the latter type.

Let S be a Sperner set of subsets of {1, 2, …, n}. (That is, for A, B ∈ S, if A ≠ B then A ⊄ B and B ⊄ A.) Then

When two triangles, ABC and A′B′C′, are perspective from a point 0, their pairs of corresponding sides meet on a line 0, the axis of perspective. The line OA passes through A′ and some point on 0. These four points have a certain cross ratio which is the same if B or C is used instead of A. The reciprocal cross ratio arises if A′B′C′ is regarded as the first triangle and ABC the second. The complete figure contains ten pairs of perspective triangles, yielding twenty cross ratios. In section 6 a technique, suggested by D. W. Babbage and John Rigby, is used to express these cross ratios in the form

Let f(n) be a real-valued additive arithmetic function, that is to say, that f(ab) = f(a) + f(b) for each pair of coprime integers a and b. Let α(x) and β(x) > 0 be real-valued functions, defined for x ≥ 2. In this paper, we study the frequencies

We shall establish necessary and sufficient conditions, subject to rather weak growth conditions upon β(x) alone, in order that these frequencies converge to the improper law, in other words, that f(n) obey a form of the weak law of large numbers.

The idea of a net will be familiar to anyone who has made a model of a three-dimensional convex polytope (3-polytope) out of a flat sheet of card or similar material. To begin with, one cuts out a polygon, and then the model is formed by folding this and joining its edges in an appropriate manner. For example, Fig. 1 illustrates two nets for a regular tetrahedron. The folds are indicated by broken lines and lettering (either of the edges or of the vertices) indicates how the edges are to be joined.

In Part I of this paper (6) we proved various index theorems for manifolds with boundary including an extension of the Hirzebruch signature theorem. We now propose to investigate the geometric and topological implications of these theorems in a variety of contexts.

It is shown that the internal and external angles at the faces of a polyhedral cone satisfy various bilinear relations. The first two of these are related to the Gauss–Bonnet and Steiner parallel formulae for spherical polytopes, while the third is completely new. However, the proofs are basically combinatorial in nature, rather than differential geometric, as in the more classical treatments. These relations lead to inversion formulae, analogous to Euler-type relations, for certain functions defined on polytopes and polyhedral cones. As a result, various new relations involving quermassintegrals and Grassmann angles are found; there is also an application to lattice polytopes.

Ramanujan's notebooks were the theme of a lecture (20) given by G. N. Watson to the London Mathematical Society in 1931. At that time, he and B. M. Wilson were collaborating on a critical edition; Watson was anticipating that the task might take a further 5 years. Unhappily, Wilson died, and the task of editing the notebooks has never been completed.

The theorem of R. Rado (12) to which I refer by the name ‘Rado's theorem for matroids’ gives necessary and sufficient conditions for a family of subsets of a finite set Y to have a transversal independent in a given matroid on Y. This theorem is of fundamental importance in both transversal theory and matroid theory (see, for example, (11)). In (3) J. Edmonds introduced and studied ‘polymatroids’ as a sort of continuous analogue of a matroid. I start this paper with a brief introduction to polymatroids, emphasizing the role of the ‘ground-set rank function’. The main result is an analogue for polymatroids of Rado's theorem for matroids, which I call not unnaturally ‘Rado's theorem for polymatroids’.

Let Rn denote Euclidean n-space. For p ∈ [0, ∞), the normalized Hausdorff p-dimensional outer measure of a set E ⊂ Rn is defined as

where the infimum is taken over all countable coverings of E by sets Ej with

and

(See, for example (4).) Then ℋP (E) measures the p-dimensional size of E. For example, ℋ0 (E) ≤ ∞ if and only if E is finite, in which case ℋ0(E) = card (E). Next if E is an arc or Jordan curve, then ℋ1(E) ≤ ∞ if and only if E is rectifiable, in which case ℋ1(E) = length (E). Finally if p is an integer in [1, n] and if E lies in a p-dimensional hyperplane T in Rn, then

where mp denotes the Lebesgue p-dimensional outer measure in T. Thus as p varies from 0 to n, the measures ℋp interpolate in a natural way between the counting measure and Lebesgue outer measure in Rn.

The object of this note is to show that elementary probability considerations suggest a very natural way of measuring the non-convexity of a set in euclidean space or, more generally, in a real Hilbert space . In particular they give a proof, much simpler and under less restrictive conditions, of results due to Shapley, Folkman and Starr which are of importance in Mathematical Economics ((1),(2)).

The ‘Radon measures’ of N. Bourbaki(1) enjoy many striking properties. Among the most important of these is the ‘strong Radon-Nikodým theorem’ that the dual of L1-(μ) can always be identified with L∞(μ) ((1), chap. 5, §5, no. 8, theorem 4). As this is certainly not true of non-σ-finite measures in general, it is natural to ask what are the special properties on which it relies.

We denote by b(n) the number of binary partitions of n, that is the number of partitions of n as the sum of powers of 2. As usual, two partitions are not considered to be distinct if they differ only in the order of their summands. It is convenient to define b(0) = 1.