A finite group is called repetition-free if its conjugacy classes have distinct sizes. It is known that a supersolvable repetition-free group is necessarily isomorphie to Sym(3). the symmetric group on three symbols. Thus the question arises as to whether Sym (3) is the only repetition-free group. In this paper it is proved that if mk denotes the minimum of the orders of the centralizers of elements of a repetition-free group G and mk ≦ 4 then G is isomorphie to Sym (3).

Using a method suggested by E. S. Barnes, it is shown that the simultaneous inequalities r(p — αr)2 < c, r(q — βr)2 < c have an infinity of integral solutions p, q, r (with r > 0), for arbitrary irrationals α and β, provided that c > 1/2.6394. This improves an earlier result of Davenport, who shows that the same conclusion holds if c > 1/46¼ = 1/2.6043 ….

Some of the conjectures of Birch and Swinnerton-Dyer have been verified for curves with complex multiplication by √ — 7. The L-function LD(1) of such curves at the point s = 1 is written as a finite sum of division values of p-functions and the integer property of LD(1) is proved.

The paper establishes a method for bounding the 2-rank of a simple group with one conjugacy class of involutions, by means of its character table. For many groups of 2-rank ≦ 4, this bound is shown to be exact. The main result is that the simple groups G2(q),(q,6) = 1, are characterized bv their character table.

Recently J. Cresp and R. P. Sullivan (1975) investigated those rings R with the following properties: (*) every multiplicative subsemigroup of R is a subring. (**) every multiplicative subsemigroup of R containing 0 is a subring. For rings with (*) they obtained the following characterization.

Several categorical aspects of localisation to compact closed categories and free compact closed categories are discussed.

A net (xy) in a vector lattice is unbounded order convergent (uo-convergent) to 0 if u ∧ |xv| order converges to 0 for all u ≧ 0. We consider, in a Banach lattice, the relationship between weak and uo-convergence. We characterise those Banach lattices in which weak convergence implies uo-convergence and those in which uo-convergence of a bounded net implies weak convergence. Finally we combine the results to characterise those Banach lattices in which weak and uo-convergence coincide for bounded nets.

An algorithm called UNLABEL is devised to uniquely label an unlabelled transitive digraph. This is used to construct a one-one correspondence between homeomorphism classes of finite nondiscrete To-topologies and certain generalised Young tableaux of shape-type α. For sets of cardinality 2, 3, 4 and 5 these classes are enumerated and classified in several ways. The notion of a generalised descriptor graph is then introduced to enumerate the homeomorphism classes of all topologies on these sets.

Suppose KG is a prime nonsingular group algebra with uniform right ideals. We show that G has no nontrivial locally finite normal subgroups. If G is soluble or residually finite, or if K has zero characteristic and G is linear, then the maximal right quotient ring of KG is simple Artinian.

A subgroup with non-trivial centre in a one-relator group is shown to be a treed HNN group (graph product) with infinite cyclic vertices. Moreover, subgroups with non-trivial centre in HNN groups are also examined.

A notion of a normal autometrized algebra is introduced which generalises the concepts of Boolean geometry. Brouwerian geometry, autometrized lattice ordered groups, semi-Brouwerian geometry, etc. The notions of ideals and congruence relations are introduced in normal autometrized algebras and a one to one correspondence between ideals and congruence relations is established. Some other common properties of the above geometries are also obtained for normal autometrized algebras.

The powerful starter-adder theorems for constructing Howell Designs are improved and consequently many types of Howell Designs that previously could only be constructed by multiplicative techniques are shown amenable to a modified starter-adder method. The existence question for Howell Designs of many new types H(s, 2n) is settled affirmatively. For prime powers pn, p ≧ 7, we reduce the entire existence question for designs of type H*(pn, 2r), pn + 1 ≦ 2r ≦ 2pn to the corresponding question for designs of type H*(p, 2m), p + 1 ≦ 2m≦2p. If these designs exist, s has no prime divisors ≤ 7 and t odd is “close” to 1, a design H * (s, s + t) is shown to exist.