This paper is concerned with the sign properties of the S-functions sλ for real arguments. We show first that sλ is indefinite if any part of the partition λ is odd. Thus it is only if all parts of λ are even that sλ can possibly be positive definite or semi-definite. In this case we show that sλ(x) is positive provided that at least l(λ) of the components of x are non-zero, where l(λ) is the number of parts of the partition λ.

The general construction of realizability toposes was first described in [3]. The data from which such a topos is constructed (called a partial applicative structure in [3]) consists of a set A equipped with a partial binary operation (called application and denoted by juxtaposition, with the convention that unbracketed expressions are evaluated from left to right) and two constants K, S satisfying

for all x, y, z ∈ A (where, as usual, an equality between possibly undefined terms means ‘one side is defined iff the other is, and then they are equal’). Following Lambek [6], we now call such a structure a partial Schönfinkel algebra, or a global Schönfinkel algebra if application is defined on the whole of A × A. (The name ‘combinatory algebra’ is also in common use.) We write for the realizability topos constructed from A.

We recall that a group G is an FC-group if for every x∈G the set of conjugates {xg|g∈G} is a finite set. Our interest here is with those groups G which are not FC groups while every proper subgroup of G is an FC-group: such groups are called minimal non-FC-groups. Locally finite minimal non-FC-groups with (G ≠ G′ are studied in [1] and the structure of these groups is reasonably well understood. In [2] Belyaev has shown that a perfect, locally finite, minimal non-FC-group is either a simple group or a p-group for some prime p. Here we make use of the results of [5] to refine the result of Belyaev and provide a positive answer to problem 5·1 of [11]; in particular, we prove the following

Theorem. There exists no simple, locally finite, minimal non-FC-group.

Let R be a Dedekind domain with field of quotients K. Let A be a finite-dimensional K-algebra. We consider isomorphism classes and genera in a category whose objects are indexed sets of full R-lattices in some ambient A-module and whose morphisms are the A-homomorphisms of the ambient A-modules which map each lattice into its corresponding lattice. We find conditions under which the stable A-isomorphism class of one particular lattice in an indexed set will determine the stable class of the indexed set within its genus. We apply our methods to show that if L/K is a tame Galois extension of algebraic number fields then the stable isomorphism class of the set of ambiguous ideals in L considered as Galois modules over K is determined by the class of the ring of integers in L together with the inertia subgroups and their standard representations over the respective residue fields of R.

Much work recently has been focused on the issue of fixed words for automorphisms of free groups - see [2] and papers referred to there. Bestvina and Handel [1] have announced a powerful structure theorem for automorphisms of free groups that has as a consequence a proof of what has become known, to the amusement of Peter Scott, as the ‘so-called’ Scott Conjecture: if F is a free group of rank n and α:F→F is an automorphism, then Fix (α) = {w∈F|α(w) = w} has rank at most n.

Let E/F be a finite normal extension of number fields with Galois group G. For each virtual character χ of G, denote by L(s, χ) = L(s, χ, F) the Artin L-series attached to χ. It is defined for Re (s) > 1 by an Euler product which is absolutely convergent, making it holomorphic in this half plane. Artin's holomorphy conjecture asserts that, if χ is a character, L(s, χ) has a continuation to the entire s-plane, analytic except possibly for-a pole at s = 1 of multiplicity equal to 〈χ, 1〉, where 1 denotes the trivial character. A well-known group-theoretic result of Brauer implies that L(s, χ) has a meromorphic continuation for all s.

Let ψ be the Grossencharacter attached to an elliptic curve E defined over an imaginary quadratic field K ⊂ of discriminant −dK, and having complex multiplication by the maximal order of K. We denote the conductor of ψ by and fix a Weierstrass model for E with coefficients in ,

whose discriminant is divisible only by primes dividing 6. Let Kab be the abelian closure of K in and choose a fundamental period Ω ∈ for the above model of the curve.

A set is called n-generic if it is Cohen generic for n-quantifier arithmetic. A (Turing) degree is n-generic if it contains an n-generic set. Our interest in this paper is the relationship between n-generic (indeed 1-generic) degrees and minimal degrees, i.e. degrees which are non-recursive and which bound no degrees intermediate between them and the recursive degree. It is known that n-generic degrees and minimal degrees have a complex relationship since Cohen forcing and Sacks forcing are mutually incompatible. The goal of this paper is to show.

Chouinard, in [8], proved the following remarkable result.

Theorem. Let k be a field of characteristic p and G be any finite group. Then a (finitely generated left) kG-module is protective if and only if it is free on restriction to all the elementary abelian p-subgroups of G.

Throughout R will denote a commutative ring with identity, A,B etc. will denote ideals of R, and E will denote a unitary R-module. We recall from [5] the definition of homological grade hgrR(A;E) as inf{r|ExtRr(R/A,E) ≠ 0}, and we allow both hgrR(A;E) = ∞ (i.e. ExtRr(R/A,E) = 0 for all r) and AE = E. For the most part E will be Noetherian, in which case hgrR(A;E) coincides with the usual grade grR(A;E) which is the supremum of the lengths of the (weak) E-sequences contained in A (see [7], for example).

The group Aut Γ of automorphisms of Rado's universal graph Γ (otherwise known as the ‘random’ graph: see [1]) and the corresponding groups Aut Γc for C a set of ‘colours’ with 2 ≤ |C| ≤ ℵ0, were studied in [4]. It was shown that Aut Γc is a simple group, and the possible cycle types of its members were classified. A natural extension of Aut Γc to a highly transitive permutation group on the same set is obtained by considering the ‘almost automorphisms’ of Γ. It is the purpose of the present paper to answer similar questions about the resulting group AAut Γc. Namely we shall classify its normal subgroups and the cycle types of its members. The main result on normal subgroups is summed up in Corollary 2·9, which says that the non-trivial normal subgroups of AAut Γc form a lattice isomorphic to the lattice of subgroups of the free Abelian group of rank n where n = |C| – 1, and for cycle types it will be shown that those occurring in AAut Γc are precisely the same as in Aut Γc except for those which are the product of finitely many cycles.

In [8] and [9] S. J. Pride has initiated a study of group presentations in which each defining relator involves exactly two members of the generating set. The methods there involve the use of graphs and so-called edge groups – the building blocks of such presentations. In this paper we replace ‘graph’ by ‘set of finite subsets of a given set’, and ‘edge group’ by ‘face group’ in order to study a larger class of presentations. This way we are able to extend to this larger class a Freiheitssatz and a result on diagrammatic asphericity which appear in the references cited above.

In this paper we investigate the following problem: is a ring R right self FP-injective if it has the property that it is pure, as a right R-module, in every ring extension? The answer is ‘almost always’; for example it is ‘yes’ when R is an algebra over a field or its additive group has no torsion. A counter-example is provided to show that the answer is ‘no’ in general. Rings which are pure in all ring extensions were studied by Sabbagh in [4] where it is pointed out that existentially closed rings have this property. Therefore every ring embeds in such a ring. We will show that the weaker notion of being weakly linearly existentially closed is equivalent to self FP-injectivity. As a consequence we obtain that any ring embeds in a self FP-injective ring.

A recurrent problem over many years in the study of linear groups has been the determination of the central height of a unipotent normal subgroup of some matrix group of specified type. In the theory of matrix groups over division rings, unipotent elements frequently present special difficulties and these have usually been by-passed by the addition of some suitable hypothesis. In this paper we make a start on the removal of these extraneous hypotheses. Our motivation for doing this now conies from [9], where by 3·7 of that paper the additional assumptions have finally reduced us to degree one, a situation where unipotent elements present few problems!

Let G be a finite group and let Irr(G) denote the set of complex irreducible characters of G. Various authors have investigated the question of how information about the degrees of the characters in Irr (G) can provide information about the structure of G. Chapter 12 of [2] gives a survey of a number of results arising from such questions. Two well-known examples of theorems that relate character degrees and group structure are those due to Thompson (12·2 in [2]) and Itô (12.34 in [2]), which we recall here.

Let q be the finite field of q elements. Denote by Sr q the projective space of dimension r over q. In Sr,q, where r ≥ 2, a k-arc is defined (see [4]) as a set of k points such that no j + 2 lie in a Sj,q, for j = 1,2,…, r−1. (For a k-arc with k > r, this last condition holds for all j when it holds for j = r−1.) A rational curve Cn of order n in Sr,q, is the set

Hamiltonian circuits with associated word an n-cycle, in Schreier right coset graphs for symmetric groups Sn mod cyclic groups Zn, correspond to change ringing principles on n bells for which the plain course is the extent; that is, neither bobs nor singles are required. This connection is made explicit for the general case, and then specialized to the cases n = 4 (minimus) and n= 5 (doubles). In particular, all 102 no-call doubles principles on three generators are found and catalogued.

For any prime number p, let Γn, p denote the congruence subgroup of SLn(ℤ) of level p, i.e. the kernel of the surjective homomorphism fp: SLn(ℤ) → SLn(p) induced by the reduction mod p (Fp is the field with p elements). We define

using upper left inclusions Γn, p ↪ Γn+1, p. Recall that the groups Γn, p are homology stable with M-coefficients, for instance if M = ℚ, ℤ[1/p], or ℤ/q with q prime and q ╪ p: Hi(Γn, p; M) ≅ Hi(Γp; M) for n ≥ 2i + 5 from [7] (but the homology stability fails if M = ℤ or ℤ/p).

It can be very difficult to determine whether or not certain rings are really full matrix rings. For example, let p be an odd prime, let H be the ring of quaternions over the integers localized at p, and set

Then T is not presented as a full matrix ring, but there is a subring W of H such that T ≅ M2(W). On the other hand, if we take H to be the ring of quaternions over the integers and form T as above, then it is not known whether T ≅ M2(W) for some ring W. The significance of p being an odd prime is that H/pH is a full 2 x 2 matrix ring, whereas H/2H is commutative. Whether or not a tiled matrix ring such as T above can be re-written as a full matrix ring depends on the sizes of the matrices involved in T and H/pH. To be precise, let H be a local integral domain with unique maximal ideal M and suppose that every one-sided ideal of H is principal. Then H/M ≅ Mk(D) for some positive integer k and division ring D. Given a positive integer n. let T be the tiled matrix ring consisting of all n x n matrices with elements of H on and below the diagonal and elements of M above the diagonal. We shall show in Theorem 2.5 that there is a ring W such that T ≅ Mn(W) if and only if n divides k. An important step in the proof is to show that certain idempotents in T/J(T) can be lifted to idempotents in T, where J(T) is the Jacobson radical of T. This technique for lifting idempotents also makes it possible to show that there are (k + n − 1)!/ k!(n−1)! isomorphism types of finitely generated indecomposable projective right T-modules (Theorem 2·10).

Suppose that G is a semi-simple, compact, connected Lie group. Endow g, its Lie algebra, with the inner product which is the negative of the Killing form. Choose a fundamental Weyl Chamber and let R+ denote the positive roots so determined.