M(22) is the sporadic simple group discovered by Fischer in 1969. The character tables of M(22) and the simple groups PSU(5, 2), PSU(6, 2), PSΩ+(6, 3) and PSΩ(7, 3) are presented. Only a brief description of the methods used to determine the tables is given.

Maximal sum-free sets in groups Zn, where n is any positive integer such, that every prime divisor of n is congruent to 1 modulo 3, are completely characterized.

Torsion topological groups with the additional property that the intersection of open sets is open are considered and their topological structure is determined:

1) the topology is uniquely determined by a normal subgroup;

2) each group is uniquely an extension of an indiscrete group by a discrete group;

3) the topology may be changed within limits without changing the dual group.

We consider the equation

εy″ + p(x)y′ + q(x)y = 0, where ε is a small positive parameter and p vanishes in the interval. Two asymptotic forms of solution are obtained and a rigorous estimate is made of the difference between the exact solutions and the asymptotic forms.

In his paper in Mat. Sb. (N.S.) 61 (103) (1963), Zareckiĭ associated with any binary relation α an ordered pair, (Lα Mα), say, of lattices and showed that α is a left [right] divisor of β if and only if We provide an alternative proof of this result by embedding the category of relations in the category of sets. Our approach provides a unified treatment of several hitherto independent results, and gives new results for the category of partial transformations.

We establish the result that a finitely generated cyclic extension of a free group is residually finite. This is done, in part, by making use of the fact that a finitely generated module over a principal ideal domain is a direct sum of cyclic modules.

We show that, under certain conditions, a soluble variety of groups which does not contain the variety of all metabelian groups is a finite exponent by nilpotent by finite exponent variety.

A calculation is made of the temperature distribution outside the laminar part of the plume produced by a small source of heat in an infinite fluid. It is shown in particular that the temperature variation is not exponentially small outside the plume, as might be expected in such a problem of boundary layer type.

There are only isolated instances of vectors of functions for which it is possible to obtain an explicit expression for the remainder functions obtained upon approximating by polynomials in the manner described by Mahler in his paper “Perfect systems”, Compositio Math. 19 (1968). We display appropriate identities and point to a pattern amongst these which suggests we should not expect convenient generalization to wider classes of functions. Proofs of perfectness do not require laborious computation but are immediate from the identities given.

It is the purpose of this paper to exhibit an infinite collection of prime, noninvertible links; each link in the collection is the union of two invertible knots. In fact, we present two such families of links. Our examples heuristically indicate that collections of links of this type exist in many varieties.

We exhibit a 3-generator metabelian group which is not finitely related but has a trivial multiplicator.

1. The purpose of this note is to establish the exitense of a finitely generated group which is not finitely related, but whose multiplecator is finitely generated. This settles negatively a question whichb has been open for a few years (it was first brought to my attention by Michel Kervaire and Joan Landman Dyer in 1964, but I believe it is somewhat older). The group is given in the follwing theorem.

Wilbur Whitten has pointed out to me that there exist invertible links amongst my family of allegedly non-invertible links [3], and a letter from Neville Smythe has confirmed that the error lies in the third paragraph from the bottom of page 106. Professor Smythe also suggested that the use of F-isotopy was possibly a red herring and, accordingly, I have devised a simpler proof of the existence of infinitely many non-invertible prime links in S3 (although Wilbur Whitten's paper [4] makes such a proof redundant).