A balanced incomplete block design or BIBD is defined as an arrangement of v objects in b blocks containing k objects all different, so that there are r blocks containing a given object and λ blocks containing any two given objects.

The nullity of the Alexander module of the fundamental group of the complement of a knot in S3 was one of the invariants of wild knot type defined and investigated by E. J. Brody in [1], in which he developed a generalised elementary divisor theory applicable to infinitely generated modules over a unique factorisation domain. Brody asked whether the nullity of a knot with one wild point was bounded above by its enclosure genus; for knots in S3, the present author showed in [6] that this was indeed the case. In [7], it was (prematurely) stated by the author that this was also the case for knots k embedded in a 3-manifold M so that H,(M — k) was torsion-free.

We consider 2π-periodic functions on the limits and give simple and complete characterizations, in terms of Fourier coefficients, of functions which belong to various Lipschitz classes and whose Fourier series are lacunary. Such characterisations seem to be missing from the literature, though there are various wellknown partial characterisations valid for functions with arbitrary spectra; cf. the remarks following Theorem 1. The results given below form complements to and sharpenings of some of the standard results valid for the special case of lacunary series.

In a survey article [1] Baumslag posed the problem of determining the abelian subgroups of a one-relator group. The solution of this problem was stated but not proved in [5], and partly solved by Moldavanskii [4]. In this paper it will be proved that the centralizer of every non-trivial element in a one-relator group with torsion is cyclic, and that the soluble subgroups of a one-relator groups with torsion are cyclic groups or the infinite dihedral group. That both types of groups may occur as subgroups is easily seen by considering

It is known (see, for instance, [1] p. 64, [6] p. 264) that, if A and B are bounded linear operators on a Banach space into itself (or, more generally, if A is a bounded linear operator on into a Banach space and B is a bounded linear operator on into), then AB and BA have the same spectrum except (possibly) for zero. In the present note, it is shown that AB is asymptotically quasi-compact if and only if BA is asymptotically quasi-compact, and that then any Fredholm determinant for AB is a Fredholm determinant for BA and vice versa.

We prove here that if F is a finitely generated free associative algebra over the field and R is an ideal of F, then F/R2 is finitely presented if and only if F/R has finite dimension. Amitsur, [1, p. 136] asked whether a finitely generated algebra which is embeddable in matrices over a commutative f algebra is necessarily finitely presented. Let R = F′, the commutator ideal of F, then [4, theorem 6], F/F′2 is embeddable and thus provides a negative answer to his question. Another such example can be found in Small [6]. We also show that there are uncountably many two generator I algebras which satisfy a polynomial identity yet are not embeddable in any algebra of n xn matrices over a commutative algebra.

Let be the free centre-by-metabelian group of rank n. In this paper, our main result is the following

Theorem. For n ≧ 4, Gn has a finite elementary abelian subgroup Hn of rank . More precisely, Hn is a minimal fully invariant subgroup contained in the centre of Gn and Gn/Hn is isomorphic to a group of 3 x 3 matrices over a finitely generated integral domain of characteristic zero.

In a recent paper [3] it was shown that the free product K * L of two fields (possibly skew) can be embedded in a field, and moreover, this latter can be chosen to be the ‘universal field of fractions’ of K*L (cf. [4, 5]). This opens up the prospect of doing for skew fields what the Neumanns and others have done for groups; indeed some sample applications were given in [3]. We pursue this topic here a little further: our main results state (i) every ccuntably generated field can be embedded in a 2-generator field, (ii) in a free product of rings over a field k, any element algebraic over k is conjugate to an element in one of the factors, (iii) any field can be embedded in a field with nth roots for each n. These results are all analogous to well known results in group theory (cf. [8]), and although the proofs are not just a translation of the group case, the latter is of scme help. Thus (ii), (iii) follow fairly easily, but they lead to other problems, still open, by replacing ‘free product of rings’ in (ii) by ‘field product of fields, and in (iii) replacing ‘nth roots’ by ‘roots of any equation’. On the other hand, (i) is less immediate, since ‘field products’ need to be used in the proof, and their manipulation requires some more technical lemmas.

Let be a class and p a property of groups. We say that p is a bigenetic property of p-groups (or more simply, p is bigenetic in p-groups) if an p-group G has the property p whenever all two-generator subgroups of G have p.

In his Canberra lectures on finite soluble groups, [3], Gaschütz observed that a Schunck class (sometimes called a saturated homomorph) is {Q, Eφ, D0}-closed but not necessarily R0closed(*). In Problem 7·8 of the notes he then asks whether every {Q, Eφ, D0}-closed class is a Schunck class. We show below with an example † that this is not the case, and then we construct a closure operation R0 satisfying Do < ro < Ro such that is a Schunck class if and only if = {QEφ, Ro}. In what follows the class of finite soluble groups is universal. Let B denote the class of primitive groups. We recall that a Schunck class is one which satisfies: (a) = Q, and (b) contains all groups G such that Q(G) ∩ B ⊆.

Many one-relator groups with center have been shown to be of the form A necessary and a sufficient condition for the sequence (P1, Q1, P2, Q2,…,Pt, Qt) are given in order for groups of the above form to be one-relator groups.

A class of groups forms a (subnormal) coalition class, or is (subnormally) coalescent, if whenever H and K are subnormal -subgroups of a group G then their join <H, K> is also a subnormal -subgroup of G. Among the known coalition classes are those of finite groups and polycylic groups (Wielandt [15]); groups with maximal condition for subgroups (Baer [1]); finitely generated nilpotent groups (Baer [2]); groups with maximal or minimal condition on subnormal subgroups (Robinson [8], Roseblade [11, 12]); minimax groups (Roseblade, unpublished); and any subjunctive class of finitely generated groups (Roseblade and Stonehewer [13]).

The four colour conjecture is well known to be equivalent to the proposition that every trivalent planar graph without an isthmus (i.e. an edge whose removal disconnects the graph) has an edge colouring in three colours ([1], p. 121). By an edge colouring we mean an assignment of colours to the edges of the graph so that no two edges of the same colour meet at a common vertex, and the graph is n-valent if n edges meet at each vertex. An edge colouring by three colours is called a Tait colouring; a trivalent graph which has a Tait colouring can be split in three edge-disjoint 1-factors, i.e. spanning monovalent subgraphs.

Let Sn denote the sphere of all points in Euclidean space Rn + 1 at a distance of 1 from the origin and Dn + 1 the ball of all points in Rn + 1 at a distance not exceeding 1 from the origin The space X is said to be aspherical if for every n ≧ 2 and every continuous mapping: f: Sn → X, there exists a continuous mapping g: Dn + 1 → X with restriction to the subspace Sn equal to f. Thus, the only homotopy group of X which might be non-zero is the fundamental group τ1(X, *) ≅ G. If X is also a cell-complex, it is called a K(G, 1). If X and Y are K(G, l)'s, then they have the same homotopy type, and consequently

Under certain general conditions an n x n stochastic matrix P = (pij), where

is known to possess the “ergodic” property. This means that there exists a finite matrix E such that

Throughout this paper a “group” will mean a “finite soluble group”.

The study of varieties of universal algebras2 which was initiated by Birkhoff in 1935, [2], has received considerable attention during the past decade; the question of particular interest being: “Which varieties have a finite basis for their laws?” In that paper Birkhoff showed that the laws of a finite algebra which involve a bounded number of variables are finitely based, so it is not altogether surprising that finite algebras have received their share of this attention.

1. In [2,5,6,7] a.o. several interpretations of the inequality for all such that were given and the following was proved.

For formations of finite soluble groups, the properties of Frattini closure and local defineability are known to be equivalent (see [2]). The investigations of Barnes and Gastineau-Hills [1] on the other hand reveal that although every Frattini closed formation of finite-dimensional soluble Lie algebras over an algebraically closed field of zero characteristic is local, without the algebraic closure condition the relationship between the two properties breaks down even for supersoluble Lie algebras. We are concerned here with the analogous problem for rings. The main results are;

Smith [6, Theorem 2.18] proved that if A is a ring which has a right artinian right quotient ring and G is a poly- (cyclic-or-finite) group, then the group ring AG has a right artinian right quotient ring. We give here a different proof (and a generalization) of this result using methods developed by Jategaonkar [3,4]. THEOREM. Let A be a ring which has a right artinian right quotient ring, and let G be a group which has a (transfinite) ascending normal series with each factor either finite or cyclic, but only a finite number of finite factors. Then AG has a right artinian right quotient ring.