In the category of locally compact groups not all families of groups have a product. Precisely which families do have a product and a description of the product is a corollary of the main theorem proved here. In the category of locally compact abelian groups a family {Gj; j ∈ J} has a product if and only if all but a finite number of the Gj are of the form Kj × Dj, where Kj is a compact group and Dj is a discrete torsion free group. Dualizing identifies the families having coproducts in the category of locally compact abelian groups and so answers a question of Z. Semadeni.

Two systems of queues are said to be isomorphic in the restricted sense if at least one of their output elements, for example waiting time, queue size, idle time, inter-departure time, and so on, has exactly or approximately the same probability distribution function, irrespective of differences in their input elements, for example, inter-arrival time, service time, and so on. This property has been used in deriving approximate solutions to systems which do not have exact solutions. The paper highlights the systems approach in queueing problems and its practical usefulness.

Groups of nilpotent length four containing a subgroup which covers and avoids the same chief factors as an -injector for some Fitting class but which is not itself an -injector have been constructed by F.P. Lockett (in his PhD thesis) and T.R. Berger and John Cossey (in preparation). Graham A. Chambers (J. Algebra 16 (1970), 442–455) has shown that such a subgroup cannot exist in a group of p–length one for all primes p. The main result of this paper closes the small gap remaining: it includes Chambers' result and establishes also that such a subgroup cannot exist in a group of nilpotent length three.

A monadicity theorem is established, for functors which satisfy the conditions of the “first isomorphism theorem” (following Linton's terminology). An application is made to the characterisation of certain types of algebraic categories generated by linear monads.

This paper deals with the study of uniform asymptotic stability of the measure differential system Dx = F(t, x) + G(t, x)Du, where the symbol D stands for the derivative in the sense of distributions. The system is viewed as a perturbed system of the ordinary differential system x' = F(t, x), where the perturbation term G(t, x)Du is impulsive and the state of the system changes suddenly at the points of discontinuity of u. It is shown, under certain conditions, that the uniform asymptotic stability property of the unperturbed system is shared by the perturbed system. To do this, the well-known Gronwall integral inequality is generalized so as to be applicable to Lebesgue-Stieltjes integrals.

It has “been open for some time whether, given an algebraic theory (triple, monad) Π in a cocomplete category K, also the category KΠ of Π-algebras must be cocomplete. We solve this in the negative by exhibiting a free algebraic theory Π in the category Gra of graphs such that GraΠ is not cocomplete. Further, we improve somewhat the well-known colimit theorem of Barr and Linton by showing that the base category need not be complete.

Let u be a nonidentity element of a finite group G and let c be a complex number. Suppose that every nonprincipal irreducible character X of G satisfies either X(l) − X(u) = c or X(u) = 0. It is shown that c is an even positive integer and all such groups with c ≤ 8 are described.

In recent years some fixed point theorems have been proved for nonexpansive mappings in Hilbert space, which have non-convex domains. The purpose of this paper is to present a simple but very useful new result of that kind and to indicate some of its consequences.

In this note, a positive solution is given to the Hahn-Banach problem for an important class of convergence vector spaces. As well, a topological vector space characterization is obtained for fully complete and Br-complete spaces.