The groups considered are Λ = GL(2, Z), π = SL(2, Z) and Θ = PSL(2, Z). A presentation of Λ is obtained for which the word problem can be solved by a simple intrinsic algorithm. The presentation is modified to display other features of Λ, and to obtain related presentations of Λ and Θ. There is an algorithm which solves the conjugacy problem of Λ.

A simple group G cannot contain a central involution t with CG(t) = 〈t〉 × M, where M is isomorphic to a simple Mathieu group.

The question whether there exist non-free projective groups of rank r in the variety has been answered in the affirmative for n ≥ 2, r ≥ 2, except for n = r = 2, by V.A. Artamonov. This paper consists in a proof that a projective group G of rank 2 in is free. If x and y are any two elements which generate G modulo , then the group F generated by x and y is free in , and the index of F in G is finite and not divisible by 2. One wishes to replace x by xu and y by yν, where u and vν lie in , so that 〈xu, yν〉 is the whole of G. This can be done: first, on general grounds, it is sufficient that 〈xu, yν〉 contain every C(a), where C(a) is the centralizer in the G/-module of an element a in (and moreover choices of u and ν for each C(a) can be combined to give a single choice good for all C(a)); second, for the particular small numbers involved, the structure of C(a) is sufficiently simple for one to pick suitable u and ν without trouble.

Consider the n-th order delay differential equation

In the last few years, the oscillatory behavior of delay differential equations has been the subject of intensive investigations. But much less is known about the equation (A) with small forcing term q(t). The only papers devoted to this problem are by Kartsatos, Kusano, and the present author. The purpose of this paper is to prove some new oscillation theorems which contain the previous results.

We introduce some generalizations of Kadec'-Klee norms and use them to study characteristics of subspaces of conjugate spaces and smoothness. We give some connections between such characteristics and basic sequences, which yield, in particular, sharpenings and simpler proofs of some known characterizations of reflexivity.

“Fuzzy theories” and “distributive laws” are used to define “fuzzy systems” in an arbitrary category. The resulting minimal realization theory provides new insights even in classical cases (so that, for non-deterministic sequential machines, the minimal realization problem is reformulated in terms of the structure of join-irreducibles in finite lattices). The definition of “fuzzy theory” is of independent interest and meshes well with philosophical aspects of fuzzy set theory.

Let λ(G) be the cardinality of a maximal sum-free set in a group G. Diananda and Yap conjectured that if G is abelian and if every prime divisor of |G| is congruent to 1 modulo 3, then λ(G) = |G|(n−1)/3n where n is the exponent of G. This conjecture has been proved to be true for elementary abelian p−groups by Rhemtulla and Street ana for groups by Yap. We now prove this conjecture for groups G = Zpq ⊕ Zp where p and q are distinct primes.

The regularity series, or briefly R-series, of a convergence space is an ordinal sequence of spaces leading to the regular modification of the space. The behavior of this series is studied relative to such basic constructs as products, subspaces, and various quotient maps. Upper bounds on the length of the R-series are obtained for several classes of spaces. This series can be employed to construct regular completions and compactifications.

Let G be a 2-transitive permutation group of a set Ω of n points and let P be a Sylow p-subgroup of G where p is a prime dividing |G|. If we restrict the lengths of the orbits of P, can we correspondingly restrict the order of P? In the previous two papers of this series we were concerned with the case in which all P–orbits have length at most p; in the second paper we looked at Sylow p–subgroups of a two point stabiliser. We showed that either P had order p, or G ≥ An, G = PSL(2, 5) with p = 2, or G = M11 of degree 12 with p = 3. In this paper we assume that P has a subgroup Q of index p and all orbits of Q have length at most p. We conclude that either P has order at most p2, or the groups are known; namely PSL(3, p) ≤ G ≤ PGL(3, p), ASL(2, p) ≤ G ≤ AGL(2, p), G = PΓL,(2, 8) with p = 3, G = M12 with p = 3, G = PGL(2, 5) with p = 2, or G ≥ An with 3p ≤ n < 2p2; all in their natural representations.

Let (E, τ) be a locally convex linear Hausdorff topological space. We have proved mainly the following results.

(i) Let f be nonexpansive on a nonempty τ-sequentially complete, τ-bounded, and starshaped subset M of E and let (I-f) map τ-bounded and τ-sequentially closed subsets of M into τ-sequentially closed subsets of M. Then f has a fixed-point in M.

(ii) Let f be nonexpansive on a nonempty, τ-sequentially compact, and starshaped subset M of E. Then f has a fixed-point in M.

(iii) Let (E, τ) be τ-quasi-complete. Let X be a nonempty, τ-bounded, τ-closed, and convex subset of E and M be a τ-compact subset of X. Let F be a commutative family of nonexpansive mappings on X having the property that for some f1 ∈ F and for each x ∈ X, τ-closure of the set

contains a point of M. Then the family F has a common fixed-point in M.

This paper finds restrictions on the parameters of supplementary difference sets in any group G with a subgroup of index 2, which therefore includes all cyclic groups of even orders. As a corollary to the main theorem, we have that if S1, …, Sr are r − {2v, k1, …, kr; 2λ} supplementary difference sets in such a group, then not all of v, k1, …, kr, λ are odd; also is the sum of r squares.

Central limit theorems are obtained for martingale arrays without the requirement of uniform asymptotic negligibility. The results obtained generalise the sufficiency part of Zolotarev's extension of the classical Lindeberg-Feller central limit theorem [V.M. Zolotarev, Theor. Probability Appl. 12 (1967), 608–618] and also the main martingale central limit theorem (not functional central limit theorem however) of D.L. McLeish [Ann. Probability 2 (1974), 620–628.

The author obtains characterizations of the quotients, epimorphisms and extreme monomorphisms in the category of separated zero-set spaces and zero-set maps (defined by Hugh Gordon [Pacific J. Math. 36 (1971), 133–157]). The method employed, that of initiality constructions, is also used to elucidate the relationship between zero-set spaces and certain other topological structures by means of forgetful functors and their right inverses. Characterizations of pseudocompactness for zero-set spaces then follow.

It has been incorrectly asserted that each non-trivial semi-simple radical class of associative rings is a variety defined by an equation of the form xn = x. In this paper we give, for each non-trivial semi-simple radical class of associative rings, a set of equations which does define that class as a variety.

The oscillatory character of the solutions of a differential equation with retarded arguments, relevant to an industrial problem, is investigated. It is also proved that one can maintain the oscillatory properties of this equation under proper conditions, if a forcing term is added to it. The results obtained extend already known results on the subject.

László Babai raised the question whether every infinite group G contains a set B of elements, of cardinal equal to the order of G, such that for elements a, b, c of B the equation a = bc−1b implies a = c. We show that the answer is affirmative for certain classes of groups, including the soluble groups and the countable groups, if the Axiom of Choice is assumed, but negative, even for abelian groups, if a different axiom, compatible with Zermelo-Fraenkel set theory but incompatible with the Axiom of Choice, is assumed.

For a Banach space X, smoothness at a point of the natural embedding ◯ in X**, is characterised by a continuity property of the support mapping from X into X*. It then becomes clear that there are many non-reflexive Banach spaces with smooth embedding, a matter of interest raised by Ivan Singer [Bull. Austral. Math. Soc. 12 (1975), 407–416].

The purpose of this paper is to obtain some common fixed point theorems for a family of mappings in a complete metric space. The results herein improve some of the recent theorems of Kiyoshi Iséki (Bull. Austral. Math. Soc. 10 (1974), 365–370).

This paper investigates the nonstandard theory of filters on a non-empty meet-semi-lattice of sets and applies this theory to the general study of topological extensions Y for a space X. In particular, we apply this theory to Baire and quasi-H-closed extensions as well as Wallman type compactifications. Whereas these extensions have previously teen obtained and studied as types of ultrafilter extensions, we study them as subsets of an enlargement of X. Since X ⊂ Y ⊂ ◯ and the elements of X and Y - X are of the same set-theoretic type, these extensions appear more natural from the nonstandard viewpoint.

In previous work, Page and Butson [Algebra Universalis 3 (1973), 112–126] characterized all equationally complete classes (atoms) of m–semigroups (universal algebras with one m–ary associative operation), and hence m–groups, in the commutative case. The further characterization of the non-commutative m-group atoms was thought to hinge upon a conjecture by Page [PhD thesis, University of Miami, 1973] that a weaker form of commutativity held true. In this paper that conjecture is proved and then used to study the special case m = 4. Two additional infinite sets of atoms are thereby determined, although it is not proved that these examples exhaust the remaining atoms for m = 4.