In the last three years a number of people have investigated the orthogonality relations for square integrable representations of non-unimodular groups, extending the known results for the unimodular case. The results are stated in the language of left (or generalized) Hilbert algebras. This paper is devoted to proving the orthogonality relations without recourse to left Hilbert algebra techniques. Our main technical tool is to realise the square integrable representation in question in a reproducing kernel Hilbert space.

I show that a group of order-automorphisms of a linearly ordered set can be expressed as an unrestricted direct product in which each factor is either the infinite cyclic group or else a group of order-automorphisms of a densely ordered set. From this a couple of simple group embedding theorems can be derived. The technique used to obtain the main result of this paper was motivated by the Erdös-Hajnal inductive classification of scattered sets.

For certain number theoretical applications, it is useful to actually compute the effectively computable constant which appears in Baker's inequality for linear forms in logarithms. In this note, we carry out such a detailed computation, obtaining bounds which are the best known and, in some respects, the best possible. We show inter alia that if the algebraic numbers α1, …, αn all lie in an algebraic number field of degree D and satisfy a certain independence condition, then for some n0(D) which is explicitly computed, the inequalities (in the standard notation)

have no solution in rational integers b1, …, bn (bn ≠ 0) of absolute value at most B, whenever n ≥ n0(D). The very favourable dependence on n is particularly useful.

Let H be a subgroup of a finite group G and let S be a set of generators of H. We prove that if G is soluble, then H is subnormal in G if and only if there exists an integer n such that for each g in G and a in S the commutator lies in H. This criterion for subnormality is also valid for soluble groups satisfying the maximal or the minimal condition on subgroups.

In an additive category A0, objects are said to be determined by their rings of endomorphisms if for each ring-isomorphism F of the rings of endomorphisms of two objects A, B in A0 there is an isomorphism f: A → B in A0 such that F(α) = fαf-1, for every endomorphism α of A. Considering.this problem in the context of closed categories (in Eilenberg and Kelly's sense), the author proves a general theorem which generalises results of Eidelheit (for real Banach spaces) and of Kasahara (for real locally convex spaces).

The following theorem is proved.

Let X be a finite set of cardinality n ≥ 2, and let F be a family of subsets of X. Suppose that for F1, F2, F3 ∈ F we have |F1 ∩ F2 ∩ F3| ≥ 2. Then |F| ≤ 2n−2with equality holding if and only if for two different elements x, y of X, F = {F ⊆ X | x ∈ F, y ∈ F}.

Let G be a finite group. A nontrivial subgroup M of G is called a CC-subgroup if M contains the centralizer in G of each of its nonidentity elements. The purpose of this paper is to classify groups with a CC-subgroup of order divisible by 3. Simple groups satisfying that condition are completely determined.

The notion of asymptotic centre of a bounded sequence of points in a uniformly convex Banach space was introduced by Edelstein in order to prove, in a quasi-constructive way, fixed point theorems for nonexpansive and similar maps.

Similar theorems have also been proved by, for example, adding a compactness hypothesis to the restrictions on the domain of the maps. In such proofs, which are generally less constructive, it may be possible to weaken the uniform convexity hypothesis.

In this paper Edelstein's technique is extended by defining a notion of asymptotic centre for an arbitrary set of nonempty bounded subsets of a metric space. It is shown that when the metric space is uniformly rotund and complete, and when the set of bounded subsets is a filter base, this filter base has a unique asymptotic centre. This fact is used to derive, in a uniform way, several fixed point theorems for nonexpansive and similar maps, both single-valued and many-valued.

Though related to known results, each of the fixed point theorems proved is either stronger than the corresponding known result, or has a compactness hypothesis replaced by the assumption of uniform convexity.

Suppose that a finite soluble group G is the product AB of subgroups A and B. Our question is the following: what conclusions can be made about G if A and B are suitably restricted? First we shall prove that the p–length of G is restricted by the derived lengths of the Sylow p–subgroups of A and B, if A and B are p–closed and p′-closed. Moreover, if in such a group the Sylow p–subgroups of A and B are modular, the p–length of G is at most 1. Next we obtain a general estimate for the derived length of the group G = AB of odd order in terms of the derived lengths of A and B. Furthermore it will be possible to bound the nilpotent length of G and also the p–length of G in terms of other invariants of special subgroups of G.

Let E be a separated, locally convex topological vector space and F a commutative family of nonexpansive mappings defined on a quasi-complete convex (not necessarily bounded) subset X of E. In this paper, it is proved that if one of the mappings in F is condensing with a bounded range then the family F has a common fixed point in X. This result improves several well-known results and supplements a recent result of E. Tarafdar (Bull. Austral. Math. Soc. 13 (1975), 241–254) for such mappings.

Let Si, i ∈ I, be a set of semigroups such that Si ∩ Sj = U, if i ≠ j, and such that U is a unitary subsemi-group of Si for each i in I. The semigroup amalgam [{Si | i ∈ I}; U] determined by this system is the partial groupoid G = USi in which a product of two elements is defined if and only if they both belong to the same Si and their product is then taken as their product in Si. In 1962, J.M. Howie showed that the amalgam G is embeddable in the free product of the Si, amalgamating U. To prove this result it suffices to find any semigroup in which G can be embedded. In this paper, by taking convenient representations of the Si, adapting a method recently (1975) used by T.E. Hall for inverse semigroups, we provide a short method of constructing such a semigroup.

A short elementary proof is given of the theorem of Marggraff which states that a primitive permutation group which contains a cycle fixing k points is (k+1)-fold transitive. It is then shown that the method of proof actually yields a generalization of Marggraff's theorem.

It is shown that a separable, quasi-reflexive Banach space of deficiency one admits an equivalent norm such that its second conjugate space is smooth; this answers a question raised by Ivan Singer [Bull. Austral. Math. Soc. 12 (1975), 407–416.

In this note we establish theorems on compactness of lattice packings.

A theorem on the existence of a solution under feasibility assumptions to a convex minimization problem over polyhedral cones in complex space is given by using the fact that the problem of solving a convex minimization program naturally leads to the consideration of the following nonlinear complementarity problem: given g: Cn → Cn, find z such that g(z) ∈ S*, z ∈ S, and Re〈g(z), z〉 = 0, where S is a polyhedral cone and S* its polar.