The recent constructions, by Rips and Olshanskii, of infinite groups with all proper subgroups of prime order, and similar ‘monsters’, show that even under the imposition of apparently very strong finiteness conditions, the structure of infinite groups can be rather weird. Thus it seems reasonable to impose the type of condition that enables us to apply the theory of finite groups. Two such conditions are local finiteness and residual finiteness, and here we are interested in the latter. Specifically, we consider residually finite groups of finite rank, where a group is said to have rank r, if all finitely generated subgroups of it can be generated by r elements. Recall that a group is said to be virtually of some property, if it has a subgroup of finite index with this property. We prove the following result:

Theorem 1. A residually finite group of finite rank is virtually locally soluble.

All groups G considered in this paper are finite and all representations of G are defined over the complex numbers. The reader unfamiliar with projective representations is referred to [6] for basic definitions and elementary results.

We recall [5] that a module T for a finite-dimensional algebra Λ is called a tilting module if

(i) T has protective dimension one;

(ii)

(iii) there is a short exact sequence 0 → Λ → T0 → T1 → 0 with T0 and T1 in add (T), the category of direct summands of direct sums of copies of T.

For arithmetical semigroups modelled on the positive integers, there is an ‘abstract prime number theorem’ (see, for example, [1]). In order to study enumeration problems in the several arithmetical categories whose prototype instead is the ring of polynomials in an indeterminate over a finite field of order q, Knopfmacher[2, 3] introduced the following modification. An additive arithmetical semigroup G is a free commutative semigroup with an identity, generated by a countable set of ‘primes’ P and admitting an integer-valued degree mapping ∂ with the properties

(i) ∂(l) = 0,∂(p) > 0 for p∈P;

(ii) ∂(ab) = ∂(a) + ∂(b) for all a, b in G;

(iii) the number of elements in G of degree n is finite. (This number will be denoted by G(n).)

The aim of this paper is to complement results by Wolfart [14] about algebraic values of the classical hypergeometric series

for rational parameters a, b, c and algebraic arguments z. Wolfart essentially determines the set of a, b, c ∈ ℚ,z ∈ ℚ for which F(a, b, c; z) ∈ ℚ and indicates, in a joint paper with F. Beukers[1], that some of these values can be expressed in terms of special values of modular forms. This method yields a few strikingly explicit identities like

but it does not give general statements about the nature of the algebraic values in question. In this paper we identify F(a, b, c; z) as a generator of a Kummer extension of a certain number field depending on z, which in particular bounds its degree as an algebraic number in terms of the degree of z. Our theorem in §2 seems to be the most precise statement one can make in general but sometimes improvements are possible as we point out at the end of §2.

In [1] B. J. Birch, solving in the affirmative a conjecture of Erdὅs, proved the following result:

Theorem 1. Let p and q be coprime integers greater than 1. Then every large natural number may be written as a sum of distinct terms of type paqb.

In fact Birch pointed out that, with similar arguments, one could obtain a stronger version where the exponent b of q can be bounded in terms of p and q. The proofs were entirely elementary.

Let K be a quadratic number field with discriminant D. The aim of this paper is to study the mean square of the Dedekind zeta function ζK on the critical line, i.e.

It was proved by Chandrasekharan and Narasimhan[1] that (1) is at most of order O(T(log T)2). As they noted at the end of their paper, it ‘would seem likely’ that (1) behaves asymptotically like a2T(log T)2, with some constant a2 depending on K. Applying a general mean value theorem for Dirichlet polynomials, one can actually prove

This may be done in just the same way as this general mean value theorem can be used to prove Ingham's classical result on the fourth power moment of the Riemann zeta function (cf. [3], chapter 5). In 1979 Heath-Brown [2] improved substantially on Ingham's result. Adapting his method to the above situation a much better result than (2) can be obtained. The following Theorem deals with a slightly more general situation. Note that ζK(s) = ζ(s)L(s, XD) where XD is a real primitive Dirichlet character modulo |D|. There is no additional difficulty in allowing x to be complex.

Let F be a non-cyclic free group, R a normal subgroup of F and G = F/R, i.e.

where π is the natural projection of F onto G, is a free presentation of G. Let R′ denote the commutator subgroup of R. The quotient F/[R′,F] is a free central extension

of the group F/R′, the latter being a free abelianized extension of G. While F/R′ is torsion-free (see, e.g. [2], p. 23), elements of finite order may occur in R′/[R′,F], the kernel of the free central extension (l·2). Since C. K. Gupta [1] discovered elements of order 2 in the free centre-by-metabelian group F/[F″,F] (i.e. (1·2) in the case R = F′), torsion in F/[R′,F] has been studied by a number of authors (see, e.g. [4–13]). Clearly the elements of finite order in F/[R′,F] form a subgroup T of the abelian group R′/[R′,F]. It will be convenient to write T additively. By a result of Kuz'min [5], any element of T has order 2 or 4. Moreover, it was pointed out in [5] that elements of order 4 may really occur. On the other hand, it has been shown in [11] that, if G has no 2-torsion, then T is an elementary abelian 2-group isomorphic to H4(G, ℤ2). So if T contains an element of order 4, then G must have 2-torsion. We also mention a result of Zerck [13], who proved that 2T is an invariant of G, i.e. it does not depend on the particular choice of the free presentation (1·1).

Let Fn be a free group of rank n and let Out Fn be its outer automorphism group. The main result of this paper is that Out F3 has a faithful representation as a group of automorphisms of the polynomial ring in seven variables over the integers. This extends a similar result for n = 2 (see Helling [3], Horowitz [5] and Rosenberger [12]), and provides a partial answer to a conjecture attributed in [5] to W. Magnus. As an application of the special nature of the representing polynomials, we obtain our second result, that Out F3 is virtually residually torsion-free nilpotent.

In [17] the second author introduced the concept of an involutary presentation. Such presentations are useful when one wants to deal geometrically with groups which have generators of order two. Now if one wants to deal with subgroups of such groups then it is no longer adequate to stick to presentations - one must consider arbitrary 2-complexes. However, these 2-complexes will differ from those which are usually considered, in that we need to allow some edges to be equal to their inverses.

We construct characteristic classes for the change of rings spectral sequences. These have their values in appropriate Ext groups and provide descriptions of the first non-zero differentials in the spectral sequences. We use these classes to do some calculations in the cohomology spectral sequence of an extension of a finite cyclic group by a finite cyclic group.

Let ℕ be the natural numbers object in , the effective topos. It was shown in [1] that the maps ℕ → ℕ are (internally or externally) just the total recursive functions. Now a subset A ⊆ ω of natural numbers corresponds to a ¬¬-closed subobject A ↣ ℕ; let kA be the least topology forcing A to be decidable, and let ℕA be the sheafification of ℕ with respect to this topology. Then one would expect the maps ℕ → ℕA to be the total functions recursive in A.

It follows from a theorem of Loday and Quillen (proposition 5·4 of [6]) that one may calculate the cyclic homology of an algebra in characteristic zero by taking a semisimplicial resolution by free algebras, quotienting out commutators and then taking homology of the resulting complex. In this paper we explain how this is a special case of a more general method based on resolutions by path algebras of directed graphs. The Loday–Quillen result may be seen as the case where the graph has only one vertex.

Let R be a Noetherian ring. If R is commutative then (a) every non-unit is contained in a height-1 prime ideal - this is the Krull principal ideal theorem - but (b) R can have an arbitrarily large stable range. The main aim of this paper is to show that if certain non-commutative rings satisfy condition (a) then they have stable range one. We shall prove the

Theorem. Let R be a Noetherian domain which is not commutative. Suppose that every non-unit of R lies in at least one height-1 prime ideal and that every height-1 prime is completely prime. Then the stable range of R is one.

In the second half of the last century the French mathematician Emil Mathieu discovered two quintuply transitive permutation groups, now labelled M12 and M24, acting on twelve and twenty-four letters respectively. With the classification of finite simple groups complete we now know that any other quintuply transitive permutation group, on any number of letters, must contain the corresponding alternating group. Indeed, the only quadruply transitive groups, other than the alternating and symmetric groups, are the point stabilizers in M12 and M24, which are denoted by M11 and M23 respectively. To put it another way, the study of multiply (≥ 4-fold) transitive groups now means the study of the symmetric groups and the Mathieu groups. Apart from their beauty and interest in their own right the Mathieu groups are involved in many of the other sporadic simple groups: see ([2], p. 238). Thus a detailed understanding of the other exceptional groups necessitates an intimate knowledge of M12 and M24.

We fix a ground field k and a finite separable extension K of k. To a Lie algebra L over k is associated the Lie algebra KL = K ⊗kL over K. If we forget the action of K, we can think of KL as a larger Lie algebra over k; in particular we can ask what is the automorphism group Autk KL of KL as a k-algebra. There does not seem to be any simple answer to this question in general; the purpose of this note is to give a simple condition on L which makes Autk KL quite easy to determine. Examples of algebras which satisfy this condition include the free nilpotent Lie algebras and the algebras of all n × n triangular nilpotent matrices.

We prove that the following conditions are equivalent: X is an F space; every ideal of C(X) is flat; every submodule of a free C(X)-module is flat.

Let G be a group. If there exists an integer n > 1 such that for each n-tuple (H1, …, Hn) of subgroups of G there is a non-identity permutation σ of Σn such that the complexes H1,…Hn and Hσ(1)…Hσ(n) are equal, then G is said to have the property of permutable subgroup products, or to be a PSP group. We show that periodic groups with this property are locally finite and investigate the structure of such groups.

A variety satisfies the amalgamation property if given G, H1, and embeddings τ1: G → H1 and τ2: G → H2, there exist and embeddings σ1: H1 → K and σ2: H2 → K such that σ1τ1 = σ2τ2.

After having investigated in [7] generic properties of compact starshaped sets in ℝd, we shall restrict here our attention to compact starshaped sets whose kernels have positive dimension. While the main results in [7] are of a topological nature and concern the whole sets, the theorems presented here describe, for kernels of codimension 0 or 1, the local aspect of the boundaries and include, for kernels of positive dimension less than d— 1, both types of statements.