In this paper, we prove that, for any prime p, a Bol loop of order 2p or of order p2 is necessarily a group, and we show that there exist exactly six non-associative Bol loops of order 8.

The packing problem for (k, 3)-caps is that of finding (m, 3)r, q, the largest size of (k, 3)-cap in the Galois space Sr, q. The problem is tackled by exploiting the interplay of finite geometries with error-correcting codes. An improved general upper bound on (m, 3)3 q and the actual value of (m, 3)3, 4 are obtained. In terms of coding theory, the methods make a useful contribution to the difficult task of establishing the existence or non-existence of linear codes with certain weight distributions.

In (l) Hunter proved that the complete symmetric functions of even order are positive definite by obtaining the inequality

where ht denotes the complete symmetric function of order t. In this note we show that the inequality can be strengthened, which, in turn, enables theorem 2 of (l) to be sharpened. We also obtain a special case of an inequality conjectured by McLeod(2).

The nilpotence class of the free commutative Moufang loop on n generators (n > 3) is the maximum allowed by the Bruck-Slaby Theorem, namely n − 1. This is proved by setting up a presentation of an extension of the loop's multiplication group as a nilpotent group of class at most 2n − 2, and then using the Macdonald-Wamsley technique of nilpotent group theory to show that this class is exactly 2n − 2.

C. A. Rogers and J. E. Jayne have asked whether, given a Polish space and an analytic subset A of which is not a Borel set, there is always a compact subset K of such that, A ∩ K is not Borel. In this paper we give both a proof, using Martin's axiom and the negation of the continuum hypothesis, of and a counter-example, using the axiom of constructibility, to the conjecture of Rogers and Jayne, which set theory with the axiom of choice is thus powerless to decide.

It has been known for some time ((6), p. 270; (4), theorem 9·19) that if is a Boolean topos, then the full subcategory Kf of Kuratowski-finite objects in is again a topos. For a non-Boolean topos , however, Kf need not be a topos, as can be seen when is the Sierpinski topos ((1), example 7·1); on the other hand, two other full subcategories of , coinciding with Kf when is Boolean, suggest themselves as candidates for a subtopos of finite objects. Of one of these, the category dKf of decidable K-finite objects in , the Main Theorem of (1) asserts that it is always a (Boolean) topos. The other is the category sKf of -subobjects of K-finite objects. The inclusions

dKf ⊆ Kf ⊆ sKf are clear.

are clear.

1. Introduction. This paper is concerned with proving identities in commutative Moufang loops. Many such identities were derived in chapter VIII of (1) in the course of demonstrating the local nilpotence of commutative Moufang loops. The results there are regarded as constituting the ‘first grammar of associators’: the reader is assumed to have a good knowledge of them. The current paper develops additional material required for the determination in (5) of the precise nilpotence class of the free commutative Moufang loop on any given finite number of generators. It is called a ‘grammar’ because it lists formal ways in which the language of associators works, and is merely meant to serve a reader of ‘literature’ in the language such as (5). However, it may be of interest for other purposes, such as answering Manin's question ((3), Vopros 10·3; (4), problem 10·2) on the 3-rank of the free commutative Moufang loop of exponent 3. There is also the problem raised below as to whether the Triple Argument Hypothesis is a consequence of the commutative Moufang loop laws. Finally, the Möbius function in Section 9 may tempt someone to look at lattice-theoretical aspects of associators.

Selmer(1) conjectured that the Hasse principle holds for all cubic surfaces of the type

that is, such a surface has a rational point whenever it has points defined over every p-adic field Qp; and he proved this assertion in the case that ab = cd.

In the paper (4) of Green and Rees it was established that the finiteness of finitely generated semigroups satisfying xr = x is equivalent to the finiteness of finitely generated groups satisfying xr−1 = 1 (Burnside's Problem). A group satisfying x2 = 1 is abelian and if it is generated by n elements, it has at most 2n elements. The free finitely generated semigroups satisfying x3 = x are thus established to be finite, and in fact the connexion with the corresponding problem for groups can be used to give an upper bound on the size of these semigroups. This is a long way from an algorithm for a solution of the word problem however, and providing such an algorithm is the purpose of the present paper. The case x = x3 is of interest since the corresponding result for x = x2 was done by Green and Rees (4) and independently by McLean(6).

1. Let V be an l-dimensional real vector space and let W be a finite subgroup of GL(V) generated by reflexions such that the space of W-invariant vectors in V is zero. Then W acts naturally on the symmetric algebra S of V preserving the natural grading . LetI be the ideal in S generated by the w-invariant elements in . The quotient algebra inherits the W-action and also a grading

is the image of Sk under S → S̅. It is well known that the W-module S̅ is isomorphic to the regular representation of W (see (3), ch. v, 5·2); in particular, S̅k = 0 for large k. (More precisely, S̅k = 0 for k > ν, where ν is the number of reflexions in W.)If ρ is an irreducible character of W, we denote by nk(ρ) the multiplicity of ρ in the W- module S̅k. The sequence n(ρ) = (n0(ρ), n1(ρ), n2(ρ), …) is an interesting invariant of the character ρ For example, in the study of unipotent classes in semisimple groups, one encounters the following question: what is the smallest k for which nk(ρ) 4= 0 (with ρ as above) and, then, what is nk(ρ) Also, the polynomial in q

can sometimes be interpreted as the dimension of an irreducible representation of a Chevalley group over the field with q elements. For these reasons it seems desirable to describe explicitly the sequence n(ρ) (or, equivalently, the polynomial Pρ(q)) for the various irreducible characters ρ of W. When W is a Weyl group of type Al, this is contained in the work of Steinberg (8); in the case where W is a Weyl group of type Bl or Dl this is done in (7),§2.

Introduction and statement of results. It is well known that if V is a vector space over a field k then its dual vector space Homk (V, k) has dimension not less than that of V, and that the dimensions are equal precisely when V is finite-dimensional. In this note we introduce a ‘dual module’ for a module over a principal ideal domain, and show that some though not all of the properties of the field case carry over.

The work of this note was motivated in the first place by North-cott's theory of dilatations for one-dimensional local rings (see, for example (4) and (5)). This produces a tree of local rings as in (4) which corresponds, in the abstract case, to the branching sequence of infinitely-near multiple points on an algebroid curve. From the algebraic point of view it seems more natural to characterize such one-dimensional local rings R by means of the set of rings which arise by blowing up all ideals Q which are primary for the maximal ideals M of R. This set of rings forms a lattice (R), ordered by inclusion, each ring S of which is a finite R-module. Moreover the length of the R-module S/R is just the reduction number of the corresponding ideal Q (cf. theorem 1 of Northcott (6)). Thus the lattice (R) provides a finer classification of the rings R than does the set of reduction numbers (cf. Kirby (1)).

A torsion-free discrete subgroup G of PSL(2, C) acts as a group of isometries of hyperbolic 3-space H3. The resulting quotient manifold M has H3 as its universal covering space with G as the group of cover transformations. We shall give examples where M has finite hyperbolic volume and is a link complement in S3. In these examples, G is a subgroup of the Picard group and in most cases is given as an HNN extension or a free product with amalgamation of kleinian groups with fuchsian groups as amalgamated or conjugated subgroups.

Let F be a field, L a commutative F-algebra and K an extension field of F. An important area of commutative algebra is the study of the passage from L to the k-algebra K ⊗FL, i.e. the investigation of the behaviour of the ideals of L under ‘extension of scalars’. In most problems of this kind one finds that the problem is reduced to the case when the algebra L is itself an extension field of F. It is for this reason that tensor products of fields play an important role (see, for example, (2), chap, viii, (3), (5), (9) and (12), vol. I).

Following P. F. Pickel (5) we write (G) for the set of isomorphism classes of finite quotients of a group G. One of the outstanding problems in the theory of polycyclic groups is to determine whether there can be infinitely many non-isomorphic polycyclic groups G with a given (G). We solve a special case of this problem with our first main result:

Theorem 1. Let G be an abelian-by-cyclic polycyclic group. Then the polycyclic-by-finite groups H withlie in only finitely many isomorphism classes.

A homomorph H is a normal Schunck class if and only if there exists a derived class χ such that H = χ*; moreover, in this case one has H′ = χ (for the definitions, see below). These results give to the derived classes a decisive significance on the study of the normal Schunck classes (see (5)). The aim of this paper is to study the homomorphs H such that H′ is a fixed derived class: we prove that these homomorphs compose a complete and distributive lattice for the inclusion relation (the maximum of this lattice being a normal Schunck class). We construct the greatest and the smallest formations whose derived class is given. We prove finally that, except in trivial cases, a normal Schunck class is not a formation.

Two recent results relate the existence of injective modules for group algebras which are ‘small’ in some sense to the structure of the group.

(1) The trivial kG-module is injective if and only if G is a locally finite group with no elements of order p = char k (9).

(2) If (G) is a countable group, then every irreducible kG-module is injective if and only if G is a locally finite p′ group which is abelian-by-finite (9) and (11)

All rings considered in this paper will be commutative and Noetherian, will have identities, and will be assumed to be non-trivial unless otherwise specified. The letter A will always denote such a ring. It is to be assumed that ring homomorphisms respect identity elements.

Let N be a von Neumann subalgebra of a von Neumann algebra M. A linear mapping π: M → N is called a retraction if it is idempotent and has norm one. By a result of Tomiyama(15) a retraction is a positive mapping and is a module homo-morphism over N. A retraction is normal if it is ultraweakly continuous, and faithful if it does not annihilate any nonzero positive element of M. Suppose that (Nn)n≥1 is an increasing sequence of von Neumann subalgebras of M whose union is weakly dense in M and that, for each n, πn: M → Nn is a faithful normal retraction. The sequence (πn) is called a martingale if, whenever m ≥ n,

In the present article we prove a result characterizing the Jordan analogues of B*-algebras among the complex Banach Jordan algebras in terms of the algebra and norm structures alone.