In this paper we will study the properties of a natural partial order which may bedefined on an arbitrary abundant semigroup: in the case of regular semigroups werecapture the order introduced by Nambooripad [24]. For abelian PP rings our order coincides with a relation introduced by Sussman [25], Abian [1, 2] and further studied by Chacron [7]. Burmistroviˇ [6] investigated Sussman's order on separative semigroups. In the abundant case his order coincides with ours: some order theoretic properties of such semigroups may be found in a paper by Burgess [5].

Let R be a commutative ring and let q be an ideal in R. Let En(R) be the subgroup of GLn(R) generated by the elementary matrices and let En(q) be the normal subgroup of En(R) generated by the q-elenientary matrices. The order of a subgroup S of GLn(R) is the ideal q0 in R generated by xij, xii−xjj, where (xij)∈S, with 1≦i, j≦n and i≠j. The subgroup S is called a standard subgroup if En(q0)≦S. An almost-normal subgroup of GLn(R) is a non-normal subgroup which is normalized by En(R).

The conference Groups–St Andrews 1985 was held at the University of St Andrews from 27 July to 10 August 1985. The conference received financial support from the Edinburgh Mathematical Society, the London Mathematical Society and the British Council. There were 366 participants from 43 countries registered for the conference. Although the conference did not specialize in a particular area of group theory, a glance at Mathematical Reviews shows that the work of the participants is mainly under classifications 20D, 20E and 20F. In part this is because the conference followed an earlier conference [6] which was primarily based on topics falling under 20F.

Throughout this paper S will denote a given monoid and R a given ring with unity. A set A is a right S-system if there is a map φ:A × S→A satisfying

and

for any element a of A and any elements s, t of S. For φ(a, s) we write as and we refer to right S-systems simply as S-systems. One has the obvious definitions of an S-subsystem, an S-homomorphism and a congruence on an S-system. The reader ispresumed to be familiar with the basic definitions concerning right R-modules over R. As with S-systems we will refer to right R-modules just as R-modules.

The Cartan matrix C of a left artinian ring A, with indecomposable projectives P1,…,Pn and corresponding simples Si=Pi/JPi, is an n×n integral matrix with entries Cij, the number of copies of the simple sj which appear as composition factors of Pi. A relationship between the invertibility of this matrix (as an integral matrix) and the finiteness of the global dimension has long been known: gl dim A < ∞⇒det C = ± 1 (Eilenberg [3]). More recently Zacharia [9] has shown that gl dim A ≦ 2⇒det C = 1, and in fact no rings of finite global dimension are known with det C = −1. The converse, det C = l⇒gl dim A < ∞, is false, as easy examples show ([[1) or [3]). However if A is left serial, gl dim A < ∞iff det C = l [1]. If A = ⊕n ≧ 0 An is ℤ-graded and the radical J = ⊕n ≧ 0 An, Wilson [8] calls such rings positively graded. Here there is a graded Cartan matrix with entries from ℤ[X] and gl dim A < ∞⇒det = 1 and, hence, det C = l [8, Prop. 2.2].

It is well-known that the number of irreducible characters of a finite group G is equal to the number of conjugate classes of G. The purpose of this article is to give some analogous properties between these basic concepts.

There are several approaches to the Stieltjes transform of generalized functions ([1, 10, 5, 6, 3, 2]). In this paper we use the definition of the distributional Stieltjes transform of index ρ (ρ ∈ ℝ\(−ℕ0); ℕ0 = ℕ∪{0}), Sρ-transform, given by Lavoine and Misra [3]. The Sρ-transform is defined for a subspace of the Schwartz space (ℝ) while in [10, 5, 6, 2] the Stieltjes transform is defined for the elements of appropriate spaces of generalized functions. In these spaces differentiation is not defined which means that the Stieltjes transform of some important distributions, for example δ(k)(x − a), a≧0, k ∈ ℕ, is meaningless in the sense of [10, 5, 6, 2]. It is easy to see that the distributions δ(k)(x − a), a≧0, k ∈ ℕ, have the Sρ-transform for ρ>−k, ρ∈ℝ\(−ℕ0). These facts favour the approach to the Stieltjes transform given in [3].

Arithmetic subgroups of reductive algebraic groups over number fields are finitely presentable, but over global function fields this is not always true. All known exceptions are “small” groups, which means that either the rank of the algebraic group or the set S of the underlying S-arithmetic ring has to be small. There exists now a complete list of all such groups which are not finitely generated, whereas we onlyhave a conjecture which groups are finitely generated but not finitely presented.

Let R be a ring with an identity and a nilpotent ideal N. Let G be a group and let R(G) be the group ring of G over R. The aim of this paper is to study the relationships between the automorphisms of G and R-linear automorphisms of R(G) which either preserve the augmentation or do so modulo the ideal N. We shall show, for example, that if G is a unique product group ([6], Chapter 13, Section 1) then every automorphism of R(G) is modulo N induced from some automorphism of G. This result, which is immediate if, for instance, R is an integral domain, is here requiring of proof since R(G) has non-trivial units (e.g. if N ≠ 0, 1 + n(g − h), ∀ n ∈ N, ∀g, h ∈ G is a unit of augmentation 1), the existence of which is responsible for some of the difficulties inherent in the present investigation. We are obliged to the referee for several helpful suggestions and, in particular, for the proof of Lemma 2.2 whose use obviates our previous combinatorial arguments.

The “other” pαqβ theorem of Burnside states the following:

Theorem A.l. Let G be a group of order pαqβ, where p and q are distinct primes. If pα>qβ, then Op(G)≠1 unless

(a) p is a Mersenne prime and q = 2;

(b) p = 2 and q is a Fermat prime; or

(c) p = 2 and q = 7.

Let G be a group and let Aut(G) be its automorphism group. It is notorious that the properties of Aut (G) do not relate well to the properties of G, perhaps the only twogeneral results being that if G has a trivial centre then the same is true of Aut (G) [2, p.89] and Baumslag's theorem that if G is finitely generated and residually finite then Aut (G) is also residually finite [1, Theorem 1, p. 117]. In the paper we shall attempt tofind analogues of these results for therelationship between the properties of R(G), the group ring of G over a ring R, and the properties of Aut R(G), the automorphism of R(G). We prove that if R(G) has a trivial centre then Aut R(G) has a trivial centre. We establish the analogue, Theorem 2.3, of Baumslag's theorem by ring-theoretic methods; our original proof used properties of group rings, the present simplified proof we owe to the referee. As an example we calculate Aut ℤ(G) in the case that G is the direct product of two cyclic groups, one of infinite order and the other of order 5. This calculation will, it is hoped, give some indication of the difficulties in determining automorphisms of the group ring of an infinite group.

In [6] Brown and Spencer noted that internal categories within the category of groups are equivalent to crossed modules. As they remarked, this result was known to various others before them, but it had not until then appeared in print. That paper led me to investigate the question of which algebraic categories, C, were such that a similar result held i.e. internal categories in C are equivalent to crossed modules of the appropriate type. The resulting work was written up in 1980 but was not submitted for publication.

In semigroup theory as in other algebraic theories a significant part of the total effort is appropriately applied to the study of certain standard examples occurring, as it were, “in nature”. The most obvious such semigroup is the full transformation semigroup (X) (see [3]) and about this semigroup a great deal is known in both the finite and infinite cases.

We study the eigenproblem

where

and Tm, Vmn are self-adjoint operators on separable Hilbert spaces Hm. We assume the Tm to be bounded below with compact resolvents, and the Vmn to be bounded and to satisfy an “ellipticity” condition. If k = 1 then ellipticity is automatic, and if each Tm is positive definite then the problem is “left definite”.

By the results of Rickman [7] and Ralston [6], a finite group G admitting a fixed point free automorphism α of order pq, where p and q are primes, is soluble. If p = q, then |G| is necessarily coprime to |α|, and it follows from Berger [1] that G has Fitting height at most 2, the composition length of <α>. The purpose of this paper is to prove a corresponding result in the case when p≠q.

In this paper we study the existence of strongly exposed points in unbounded closed and convex subsets of the positive cone of ordered Banach spaces and we prove the following characterization for the space l1(Γ): A Banach lattice X is order-isomorphic to l1(Γ) iff X has the Schur property and X* has quasi-interior positive elements.

The idea of a pregroup was introduced by Stallings and provides an axiomatic setting for a well-known argument, due to van der Waerden, used to prove normal form theorems. Details are provided in [7], Section 3.

This paper is a continuation of our project on “inverse interpolation”, begun in [6]. In brief, the task of inverse interpolation is to deduce some property of a function f from some given property of the set L of its Lagrange interpolants. In the present work, the property of L is that it be a uniformly bounded set of functions when restricted to the domain of f. In particular (see Section 3), when the domain is a disc, we deduce sharp bounds on the successive derivatives of f. As a result, f must extend to be an analytic function (of restricted growth) in the concentric disc of thrice the original radius.

Let G be a group and K a field. We denote by (KG) the group of units of the group ring of G over K and for a group X we denote by T(X) the setof torsion elements of G i.e., the set of all elements of finite order.

Let E be n-dimensional (n≧2) real vector space with a nondegenerate symmetric scalar product (.|.):E × E → R1 with an arbitrary signature (p, n–p). Let us consider a second order partial differential equation (P.D.E.) of the form:

where φ is a given function of two variables, v is an unknown function (defined on an open subset 0 ⊂E), |∇ν|2: =(∇ν|∇ν) is the square of the gradient ∇ν of the function ν and ∇2, denotes the Laplace-Beltrami operator.