1. Let X be a closed Riemann surface of genus g ≥ 2 and let Aut X denote the group of automorphisms of X where, in this paper, an automorphism means a conformal or anticonformal self-homeomorphism. X is called hyperelliptic if it admits a conformal automorphism J of order 2 such that X/H has genus 0, where H = 〈J〉 is the group of order 2 generated by J. Thus X is a two-sheeted covering of the sphere which is branched over 2g + 2 points and J is the sheet-interchange map. J is the unique conformal automorphism of order 2 such that X/〈J〉 has genus 0 and it follows that if U ∈ Aut X, then UJU−1 = J. Thus J is central in Aut X and H ≤ Aut X. (Cf. [8])

We present an elementary proof of the theorem, usually attributed to Noether, that if L/K is a tame finite Galois extension of local fields, then is a free -module where Γ=Gal(L/K. The attribution to Noether is slightly misleading as she only states and proves the result in the case where the residual characteristic of K does not divide the order of Γ [4]. In this case is a maximal order in KΓ which is not true for general groups Γ. There is an elegant proof in the standard reference [2], but this relies on a difficult result in representation theory due to Swan. Our proof depends on a close examination of the structure of tame local extensions, and uses only elementary facts about local fields. It also gives an explicit construction of a generator element, and the same proof works both for localizations of number fields and of global function fields.

In an earlier paper [1] on groups which are the products of two finite cyclic groups with trivial intersection, certain permutations, called “semi-special”, played a certain role. The permutation π of the numbers 1, 2,…, n is semi-special if† πn=n, and if, for every y ε [n],

is again a permutation, namely a power (depending on y) of π.

This paper considers the determination of the coefficients in two sets of triple trigonometrical series and shows that these can be obtained in closed form. The series considered are special cases of some triple series in Jacobi polynomials studied by K. N. Srivastava [1]. Srivastava, however, shows that the problem for the more general series can be reduced to the solution of a Fredholm integral equation of the second kind and he does not discuss special cases which may lead to closed form solutions.

If X is a class of groups, the class of counter-Xgroups is defined to consist of all groups having no non-trivial X-quotients. The counter-abelian groups are the perfect groups and the counter-counter-abelian groups are the imperfect groups studied by Berrick and Robinson [2]. This paper is concerned with the class of counter-counterfinite groups. It turns out that these are the groups in which any non-trivial quotient has a non-trivial representation over any finitely generated domain (Theorem 1.1), so we shall call these groups highly representable or HR-groups.

The Eckman–Hilton duality [4] reverses arrows in diagrams, turns products to co-products, and multiplications to co-multiplications, etc. In accordance with this process, Kan [5] obtained the dual of a monoid structure in the category of groups. In this way, we obtain co-monoid structures on topological groups. The main result of this paper is that for kaω groups (see §2), we obtain a one-to-one correspondence between the co-monoid structures, and the free topological bases of the group (§3), thus obtaining topological analogues of the main results of [5].

In [2] a condition, originally due to Olagunju, was given for the spectra of certain compact operators to be on the real axis of the complex plane. Here, by using conformal mappings, this result is extended to more general curves. The problem divides naturally into two cases depending on whether or not the curve under consideration passes through the origin. Discussion is confined to the prototype curves C0 and C1. The case of C0, the unit circle of centre the origin, is considered in § 3; this problem is a simple one as the spectrum is a finite set. In § 4 results are given for C1 the unit circle of centre the point 1, and some results on ideals of compact operators, given in § 2, are needed. No attempt has been made to state results in complete generality (see [2]); this paper is kept within the framework of Hilbert space, and particularly simple conditions may be given if the operators are normal.

Given a group presentation (or more generally† a 2-complex) one can associate with it an object which has variously been called the co-initial graph, star-graph, star-complex, and which has proved useful in several contexts [2], [6], [7], [8], [9], [10], [12]. For certain mappings of 2-complexes φ: ⃗ℒ (”strong mappings”) one gets an induced mapping φst: st⃗ℒst of the associated star-complexes. Then st is a covariant functor from the category of 2-complexes (where the morphisms are strong mappings) to the category of 1-complexes, and this functor behaves very nicely with respect to coverings (Theorem 1).

The relation ℒ* is defined on a semigroup S by the rule that a ℒ*b if and only if the elements a, b of S are related by Green's relation ℒ in some oversemigroup of S. A semigroup S is an E-semigroup if its set E(S) of idempotents is a subsemilattice of S. A right adequate semigroup is an E-semigroup in which every ℒ*-class contains an idempotent. It is easy to see that, in fact, each ℒ*-class of a right adequate semigroup contains a unique idempotent [8]. We denote the idempotent in the ℒ*-class of a by a*. Then we may regard a right adequate semigroup as an algebra with a binary operation of multiplication and a unary operation *. We will refer to such algebras as *-semigroups. In [10], it is observed that viewed in this way the class of right adequate semigroups is a quasi-variety.

In [4], Ramanujan stated the following beautiful identity which was later proved by Bailey [2] and [3];

We denote by S(n) the coefficient of qn on the right hand side of (1). The object of this paper is to prove the following two theorems.

We show how the solution to certain diophantine equations involving the discriminant of complex quadratic fields leads to the divisibility of the class numbers of the underlying fields. This not only generalizes certain results in the literature such as [2], [4]–[6] but also shows why certain hypotheses made in these results are actually unnecessary since, as our criteria demonstrate, these hypotheses are forced by the solution of the diophantine equations involved. Our methods are based only on the most elementary properties of a principal ideal in a complex quadratic field.

For a completely 0-simple semigroup, Howie [2] has investigated the subsemigroup generated by the idempotents. Here we determine those elements of such a semigroup which are generated by the set of nilpotent elements and hence we derive a condition for a completely 0-simple semigroup to be nilpotent generated. This condition is purely combinatorial, in terms of the structure of the graph associated with the semigroup, and it includes the case of a non-regular Rees matrix semigroup.

In this paper, we present an “order” characterization of completely bounded bimodule maps for bimodules over unital operator algebras. We use this result to prove a bimodule generalization of Wittstock's generalized Hahn-Banach theorem. Our proofs simplify and unify some of Wittstock's arguments.

For a single space curve (that is, a smooth curve embedded in ℝ3) much geometrical information is contained in the dual and the focal set of the curve. These are both (singular) surfaces in ℝ3, the dual being a model of the set of all tangent planes to the curve, and the focal set being the locus of centres of spheres having at least 3-point contact with the curve. The local structures of the dual and the focal set are (for a generic curve) determined by viewing them as (respectively) the discriminant of a family derived from the height functions on the curve, and the bifurcation set of the family of distance-squared functions on the curve. For details of this see for example [6, pp. 123–8].

An inverse semigroup S shall be said to be harmonic if every congruence on S is determined by any one of its classes. In other words, if λ and ρ are congruences on S having a congruence class in common, then λ = ρ. The class of all harmonic semigroups contains all bisimple inverse semigroups, as proved by Žitomirskiĭ [11] and also by Schein [10], and all congruence-free inverse semigroups. Moreover, is contained in the class of all 0-simple or simple inverse semigroups, as is easy to see. We shall show that there exist non-bisimple, non-congruence-free harmonic semigroups and that there are simple inverse semigroups which are not harmonic.

A compact bordered Klein surface X of algebraic genus g ≥ 2 has maximal symmetry [6] if its automorphism group A(X) is of order 12(g — 1), the largest possible. The bordered surfaces with maximal symmetry are clearly of special interest and have been studied in several recent papers ([6] and [9] among others).

Green's relations are essential for “co-ordinatizing” semigroups. Jacqueline Klasa, in applying cognate ideas to categories [4, 5], has shown that divisibility in suitably-behaved categories may be described in terms of subobjects and quotients.

Here it is shown that adjoint functors which are onto objects preserve divisibility (in a certain sense). The inclusion functor of the category of sets into the category R of binary relations is such a functor. A slight modification of its right adjoint allows the representation of R as a full subcategory in a category CSL of complete semilattice morphisms.