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).

Suppose U and V are to m × n matrices over the complex field. We obtain a representation for the Moore–Penrose inverse of the sum U + V. A well-known result of Cline is then derived as a special case of a corollary of this representation.

This is the text and invited lecture given at the 20th Annual Meeting of the Australian Mathematical Society in Perth, May 1976.

In this talk I want to give a survey of some of the areas of research in ordinary differential equations. In view of the vastness of the field, the limited thime available and may own inadequacies I will restrict attention to problems in which I have been involved in some way, although I won't be speaking about my own work. Even with this restriction it will be necessary to be somewhat superficial. However, in spite of all these drawbacks, I thought it would assist communication among mathematicians simply to have an idea of what other people are doing. Since this is my purpose I will not mention the quite mild smoothness hypotheses in the statements of some results.

Let Lp(Ω, K) denote the Banach space of weakly measurable functions F defined on a finite measure space and taking values in a separable Hilbert space K for which ∥ F ∥p = ( ∫ | F(ω) |p)1/p < + ∞. The bounded Hermitian operators on Lp(Ω, K) (in the sense of Lumer) are shown to be of the form , where B(ω) is a uniformly bounded Hermitian operator valued function on K. This extends the result known for classical Lp spaces. Further, this characterization is utilized to obtain a new proof of Cambern's theorem describing the surjective isometries of Lp(Ω, K). In addition, it is shown that every adjoint abelian operator on Lp(Ω, K) is scalar.

Necessary and sufficient sonditions are given for the existence of the group and Drazin inverses of bidiagonal and triangular Toeplitz matrices over an arbitrary ring.

This paper contains proofs of the compactness, completeness and Löwenheim-Skolem theorems for predicate logic, together with their application to nonstandard numbers; proofs of undecidability in predicate logic and number theory, and the Gödel incompleteness theorem.

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 ….

If T = (tij) is an n x n matrix with non-negative integer entries, we define a transformation T: Cn → Cn by z' = Tz where

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A sufficient condition, which may also be necessary, for the regularity of the direct product of two finite regular p-groups is obtained. In an attempt to decide the necessity of the condition, we look at two related questions. Firstly, does the regularity of the direct square of a regular p-group imply the regularity of all the finite direct powers; secondly, is the join of two regular varieties regular? Fainally, the results obtained are used to give some precise answers in special cases.

In this paper the concept of length as defined for groups by Lausch–Nöbauer in their book Algebra of Polynomials (North Holland, Amsterdam, 1973) is generalized in several ways. It turns out that the main results of Lausch-Nöbauer concerning it remain valid for this generalization.

The now well-known Coale–Lopez theorem, and variants of it, state that, under certain conditions, the product of the first r members of an infinite inhomogeneous sequence of nonnegative matrices approaches, as r → ∞, the class of matrices which, apart from scalar multiples, have only one distinct column. The aim of the present paper is to lay down conditions under which such products approach the class of matrices which, apart from scalar multiples, have no more than d distinct columns. A stronger result is then obtained by considering stochastic matrices instead of just nonnegative ones.

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.

We show that if G is a connected graph of order n such that no line lies in more than one cycle (in other words, G is a cactus of order n), then the stability index of G is one of the integers 0, 1, n−7, n−6, n−5, n−4 or n.

The set of bounded elements of a unital l.m.c. algebra is characterised as the union of certain naturally defined normed subalgebras, and an analogous characterisation is given for algebras of quotient-bounded operators on a locally convex space. Pseudocomplete l.m.c. algebras are characterised in terms of the completeness of these subalgebras, and the equivalence of this condition with the pseudocompleteness of the quotient-bounded operator algebras established. The scalar multiples of the identity in a unital l.m.c. algebra are characterised in terms of certain boundedness conditions.

In this note, a one sided aspect of regular variation is considered, and some different results which can be obtained by bounding the ratio f(xλ)/f(x) above or below are given.

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.

It is shown that (n being an integer) any non-trivial finite linear space with n2- 1 points, all of degree at most n+1, is embeddable in a finite projective plane of order n. This generalizes a theorem of Bose and Shrikhande and settles the unsolved case n = 6.

If h is a homomorphism of an N-semigroup onto an N-semigroup S′, then h induces a homomorphism h of the translation semigroup Λ(S) of S into Λ(S′) of S′. We will study the relations between the structures of S, S′ and Λ(S), Λ(S)′, and will introduce the specialized concepts of N-semigroups S. In particular, we will be interested in power-joined steady or endless N-semigroups. Finally, we will consider admissibility of torsion abelian groups, that is, consider what torsion abelian group can be a structure group of a power-joined steady N-semigroup.

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.

The problem considered is that of approximating irrationals α by rationals p/q where p and q avoid certain congruence classes mod 2k for certain integers k. Results are obtained which give close bounds on a number c such that |α - p/q| < c/q2 has infinitely many solutions where p and q can be expressed as the sum of three squares.