It is a trivial consequence of Magnus' solution to the word problem for one-relator groups [9] and the existence of finitely presented groups with unsolvable word problem [4] that not every finitely presented group can be embedded in a one-relator group. We modify a construction of Aanderaa [1] to show that any finitely presented group can be embedded in a group with twenty-six defining relations. It then follows from the well-known theorem of Higman [7] that there is a fixed group with twenty-six defining relations in which every recursively presented group is embedded.

Let Bd(e) denote the Burnside group with d ≥ 2 generators a1,.a2, …, a, and exponent e > 0, i.e., the free group of rank d of the Burnside variety of exponent e. It is known that Bd(e) is infinite for all sufficiently large odd values of e; cf. Novikov and Adyan [3] or Britton [1]. In particular Bd(pk), where p is an odd prime, is infinite for all sufficiently large k. It is not known whether or not Bd(2k) is infinite for all sufficiently large k infiniteness would imply that Bd(n) is infinite for all sufficiently large n, as has been conjectured by Novikov [2].

It is well known (see Latyšev [1] for finite dimensional case) that the universal envelope of Lie Alaebra g over a commutative field ť of characteristic 0 is a PI-alaebra (i.e. possesses a nontrivial identity) if and only if this Lie algebra is abelian. On the other hand the recent results due to Passman [2] describe the conditions under which the group algebra of a group over an arbitrary commutative field is a PI-algebra. A. L. Šmel'kin suggested that I should find necessary and sufficient conditions for a Lie algebra g over a field of nonzero characteristic under which its universal envelope Ug should be a PI-algebra. These conditions are given in the following theorem.

A system or family (Aγ: γ∈ N) of sets Aγ, indexed by the elements of a set N, is called an (a, b)-system if ¦N¦ = a and ¦Aγ¦ = b for γ ∈ N. Expressions such as “(a, <b)-system” are self-explanatory. The system (Aγ: γ∈N) is called a δ-system [1] if Aμ∩Aγ = Ap ∩ Aσ whenever μ, γ, ρ, σ ∈ N; μ≠ γ; ρ ≠ σ. If we want to indicate the cardinality ¦N¦ of the index set N then we speak of a δ(¦N¦) system. In [1] conditions on cardinals a, b, c were obtained which imply that every (a, b)-system contains a δ(c)-subsystem. In [2], for every choice of cardinals b, c such that the least cardinal a = fδ(b, c) was determined which has the property that every (a, < b)-system contains a δ(c)-subsystem.

The following theorem is the focal point of the present paper. It stipulates an algebraic condition equivalent, in any finitely generated group, to the solubility of the word problem.

THEOREM I. A necessary and sufficient condition that a finitely generated group G have a soluble word problem is that there exist a simple group H, and a finitely presented group K, such that G is a subgroup of H, and H is a subgroup of K.

This paper, the second of a sequence beginning with [14], deals with the relationship between a distributive lattice L = (L; ∨, ∧, 0) with zero, and certain spaces of minimal prime ideals of L. Similar studies of minimal prime ideals in commutative semigroups [8] and in commutative rings [6] inspired this work, and many of our results are similar to ones in these two articles. However the nature of our situation enables many of these results to be pushed deeper and thus to arrive at a more satisfactory state; indeed with the insight obtained from the simpler lattice situation, one can return to some topics considered in [6], [8] and give complete accounts. We do not do this in the present paper, but leave the details to the reader, see e.g. [15]. Also a study of minimal prime ideals illuminates some topics in the theory of distributive lattices, particularly Stone lattices.

We introduce further finite groups which can be presented with an equal number of generators and relations.

In [4] a concept of a weakly projectable vector lattice has been introduced. Stone vector lattices [3] and thus all special types of them, like Riesz [5], σ-complete and complete vector lattices are weakly projectable. Moreover C[0, 1] is weakly projectable but not Stone [4]. As we see the collection W of weakly projectable vector lattices is quite large. This explains to some extent the difficulty in producing examples of vector lattices which do not belong to W. In this note an example of a Banach lattice [1] which is not weakly projectable is described.

It is well known that a wide range of Special Function Theory can be realized by considering unitary representations of certain topological groups.

In this approach it is very important to determine all irreducible continuous unitary representations of the group in question.

For the group of movements this problem was initiated by Vilenkin [6]. Rather restrictive conditions were imposed in this paper and while he returned to the problem in [7], it was still not solved in full generality (among other things the representation space was assumed separable). The first complete solution appears to have been given by Thoma [5]. Here, the method was to show each irreducible continuous unitary representation equivalent to a particular representation in a space of square integrable functions.

In [1], Owen gave sufficient conditions for the uniqueness of certain mixed problems having elliptic and hyperbolic nature for the ultrahyperbolic equation. Recently, Diaz and Young [2] has obtained necessary and sufficient conditions for the uniqueness of solutions of the Dirichlet and Neumann problems involving the more general ultrahyperbolic equation The purpose of this paper is to present corresponding uniqueness conditions for the Dirichlet and Neumann problems for the singular ultrahyperbolic equation for all values of the parameter, α −∞ > α > ∞. The symbol Δ denotes the Laplace operator in the variables x1, …, xm, Dj indicates partial differentiation with respect to the variable yj (1 ≧ J ≧ n), and the summation convention is adopted for repeated indices including (iu2), where denotes differentiation with respect to the variable xi.

Birkhoff and Pierce [2] introduced the concept of an ƒ-ring and showed that an l-ring is an f-ring if and only if it is a subdirect product of totallyordered rings. An l-ideal of an f-ring R is an algebraic ideal which is at the same time a lattice ideal of R. Structure spaces (i.e. sets of prime ideals endowed with the so-called hull-kernel or Stone topology) for ordinary rings have been studied by many authors. In this paper we consider certain analogues for ƒ-rings, and give characterisations of ƒ-rings for which these structure spaces are discrete.

This note is a continuation of the author's work [6], describing the structure of a finite group given some information about the distribution of the subnormal subgroups in the lattice of all subgoups.

DEFINITION. An upper chain of length n in the finite group G is a sequence of subgroups of G; G = Go > G1 > … > Gn, such that for each i, Gi is a maximal subgroup of Gi-1. Let h(G) = n if every upper chain in G of length n contains a proper ( ≠ G) subnormal entry, and there is at least one upper chain in G of length (n – 1) which contains no proper subnormal entry.

Let R be the set of real numbers, and let S1 denote the class of all real valued functions f on R which are smooth to the first order (i.e. the derivative ƒ(1) exists and is continuous) and have compact support. The first order variation off on an open set U is given by and in the case where U = R we have the total first order variation of f, usually denoted by I1(f).