Let K be an algebraic number field. If q is a prime ideal of the ring of integers of K and α is a number of K prime to q then Mq(α) denotes the multiplicative group generated by α modulo q. In the paper [5] there is the remark: ‘We do not know whether for all a, b, c ∈ ℚ with abc ≠ 0, |a| ≠ 1,|b| ≠ 1,|c| ≠ 1 there exist infinitely many primes q with Mq (a) = Mq (b) = Mq (c).’

We find the adjoint of the Askey–Wilson divided difference operator with respect to the inner product on L2(–1, 1, (1– x2)½dx) defined as a Cauchy principal value and show that the Askey-Wilson polynomials are solutions of a q-Sturm–Liouville problem. From these facts we deduce various properties of the polynomials in a simple and straightforward way. We also provide an operator theoretic description of the Askey-Wilson operator.

The theorem which is still known as Nakayama's Conjecture shows how the modular characters of the symmetric group Sn can be divided into blocks of various weights, those with the same weight having similar properties. In fact, all blocks of weight one have essentially the same decomposition numbers and these are easy to describe. In recent work, Scopes [16, 17] has shown that there are essentially only finitely many possibilities for the decomposition numbers for blocks of any given weight, and has given a bound for the number. We develop the combinatorics implicit in her work, and so establish an improved bound.

The role played by the classical braid groups in the interplay between geometry, algebra and topology (see [Ca]) derives, in part, from their definition as the fundamental groups of configuration spaces of points in the plane. Seeking to generalize these groups and to understand them better, one is led to ask: are there other discrete groups whose topological invariants arise from configuration spaces?

The groups of marked homeomorphisms (1·1) provide a positive response which is in some sense banal; the realization problem (1·5) is to find non-banal examples.

The Harada-Norton group is one of the twenty-six sporadic simple groups. It has order 273, 030, 912, 000, 000 = 214.36.56.7.11.19. In this paper our main objective is:

Theorem 1. The Harada-Norton group acts as a group of linear automorphisms of a 133-dimensional commutative, non-associative algebra defined over F5.

A positive integer is square-full if each prime factor occurs to the second power or higher. Each square-full number can be written uniquely as a square times the cube of a square-free number. The perfect squares make up more than three-quarters of the sequence {si} of square-full numbers, so that a pair of consecutive square-full numbers is a pair of consecutive squares at least half the time, with

Ankeny–Artin–Chowla obtained several congruences for the class number hk of a quadratic field K, some of which were also obtained by Kiselev. In particular, if the discriminant of K is a prime number p ≡ 1 (mod 4) and ε = t + u √p/2 is the fundamental unit of K, then

Let Γ be a discrete group. Γ is said to have finite virtual cohomological dimension (vcd) if there exists a finite index torsion-free subgroup Γ′ of G such that Γ′ has finite cohomological dimension over ℤ. Examples of such groups include finite groups, fundamental group of a finite graph of finite groups, arithmetic groups, mapping class groups and outer automorphism groups of free groups. One of the fundamental problems in topology is to understand the cohomology of these finite vcd-groups.

In this paper we conduct a systematic, computerized search for groups generated by small, but highly symmetric, sets of involutions. Many classical groups are readily obtained in this way, as are a number of sporadic simple groups. The techniques of symmetric generation developed elsewhere are described afresh, and the results are presented in a convenient tabular form.

Let N/K be a tame, abelian extension of number fields, whose Galois group is denoted by Γ. The basic object of study in this paper is the ring of integers of N, endowed with the trace form TN/K; the pair is then a Hermitian module (where we abbreviate ), and it restricts to a -Hermitian module (The reader is referred to Section 2 for the basics on Hermitian modules.) Ideally one would like to determine completely the class of this Hermitian module in K0H(ℤΓ), the Grothendieck group of ℤΓ-Hermitian modules modulo orthogonal sums; however, in general when Γ is even, one knows that even the ℚΓ-Hermitian module given by restricting (N, TN/K) is difficult to classify. (See for instance [S] and [F2].) To circumvent this difficulty we proceed in the following fashion, as suggested by the recent work of P. Lawrence (see [L]): let D = D(Γ) denote the anti-diagonal of Γ in Γ × Γ, that is to say

and let N(2) = (N⊗KN)D so that N(2) is a Galois algebra over K with Galois group Γ × Γ/D(Γ) ≅ Γ. Write for the trace form of N(2)/K, and define the -order it is then easy to see that is isomorphic as an -module to (see (3·1·6)). To be precise we ought really to write etc.; however, the base field will always be clear from the context.

Let ACCR(L) be the CCR algebra over a Hubert space L, and let {σt}t∈R be a Bogoliubov automorphism group of ACCR(L) induced by a strongly continuous one-parameter unitary group on L. In this paper, we introduce some continuity for linear functionals on ACCR(L) and, under this continuity, we study existence, uniqueness, and non-uniqueness of ground states for {ACCR(L), {σt}t∈R}.

The Scott Conjecture, proven by Bestvina and Handel [2] says that an automorphism of a free group of rank n has fixed subgroup of rank at most n. We characterise in Theorem A below those automorphisms that realise this maximum. It follows from this characterisation, for example, that any such automorphism has linear growth. In our paper [3], we generalised the Scott Conjecture to arbitrary free products, using Kuros rank (see Section 2 below) in place of free rank; in Theorem B, we characterise those automorphisms of a free product realising the maximum. We show that in this case the growth rate is also linear. These results extend those of [4].

Several results are obtained relating to the modular representation theory of the general linear group GLn in the defining characteristic p > 0. In Section 1, embeddings of certain simple modules in symmetric powers of the natural module, or in tensor products of truncated symmetric powers, are constructed. In Section 2, cases are found where simple quotientsof Schur modules H0(λ) can be constructed by extending theidea of truncation to these modules in a natural way. In Section 3, the characters of those simple modules which can be constructed as twisted tensor products of truncated symmetric powers are expressed in terms of Schur functions.

The intuition behind the notion of a cycle-free partial order (CFPO) is that it should be a partial ordering (X, ≤ ) in which for any sequence of points (x0, x1;…, xn–1) with n ≤ 4 such that xi is comparable with xi+1 for each i (indices taken modulo n) there are i and j with j ╪ i, i + 1 such that xj lies between xi and xi+1. As its turn out however this fails to capture the intended class, and a more involved definition, in terms of the ‘Dedekind–MacNeille completion’ of X was given by Warren[5]. An alternative definition involving the idea of a betweenness relation was proposed by P. M. Neumann [1]. It is the purpose of this paper to clarify the connections between these definitions, and indeed between the ideas of semi-linear order (or ‘tree’), CFPO, and the betweenness relations described in [1]. In addition I shall tackle the issue of the axiomatizability of the class of CFPOs.

While this paper is principally a continuation of [5], with as its object the application of sections 6 and 7 of that paper to obtain results related to the Buchsbaum–Rim multiplicity, it also has connections with [8] which are the subject of the first of the four sections. These concern integral equivalence of finitely generated R-modules. where R is an arbitrary noetherian ring. We therefore introduce a finitely generated R-module M and relate to it a short exact sequence (s.e.s.),

where F is a free module generated by m elements u1,…, um, and L is generated by elements yj, (j = 1, …, n), of F. We identify the elements u1, …, um with a set of indeterminates X1, …, Xm, and F with the R-module S1 of elements of degree 1 of the graded ring S = R[X1, …, Xm].

Extra special p-groups are groups which are central extensions of ℤ/p by elementary abelian p-groups. The cohomology ring of these groups occupies an important place in equivariant cohomology theories and in representation theories. Quillen[6] decided the cohomology for p = 2 and Tezuka-Yagita [7] studied the varieties defined by its mod p cohomology for odd prime p. However, for applications we need more information about the behaviour of the Hochschild-Serre spectral sequence (see [3] p. 295, [1] p. 6).

Topological circle actions on 4-manifolds are studied using modifications of known techniques for smooth actions. This yields topological versions of some previously known restrictions on the fundamental groups of 4-manifolds admitting smooth circle actions.

Let Bn denote the Artin braid group on ‘n-strings[ and PBn its normal subgroup consisting of all the pure braids [Bi, Mo]. These groups have been considerably scrutinized by both topologists and algebraists [BL]. One question whose answer has so far eluded us is whether or not the Gassner representation G: PBn → Mn × n(λ), into the group of n-by-n matrices over , is faithful (see Section 1) [Bi; ·3] [Ga]. Recently the less discriminating Burau representation B: PBn → Mn × n(Z[t±1] ) was shown to have a non-trivial kernel for each n ≥ 6 [M, LP] but these techniques have not yet yielded an element of kernel(G). This paper is a partial step in that direction.

Let P be a fixed set of primes, the category of all groups and group-homomorphisms, and the full subcategory of nilpotent groups. In [9], an idempotent functor called P-localization, was defined so as to extend the ℤ-module-theoretic localization of abelian groups. There are two well-known extensions of e to , namely, Bousfield's P-localization [2], [4], denoted by EZP, and Ribenboim's P-localization [13], usually denoted by ( )P. Ribenboim's P-localization is the maximal extension among localizations extending e to in that it maximizes the number of groups in its image [7]. The localized groups obtained after applying Ribenboim's P-localization are precisely the P-local groups, that is, the groups having unique nth-root for every n whichis co-prime to P, [13]. Being maximal is equivalent to this class of P-localized groups being the saturated class of groups generated by e–equivalences, that is, group homomorphisms between nilpotent groups which become isomorphisms after applying e.

Traced monoidal categories are introduced, a structure theorem is proved for them, and an example is provided where the structure theorem has application.