In (1), Carter and Hawkes define the -normalizers of a finite soluble group for any saturated formation . These subgroups are conjugate, invariant under homomorphisms of the group, cover and avoid the chief factors of the group and may be characterized by means of the maximal chains of subgroups connecting them to the group. The first aim of the present paper is to generalize both the -normalizers and the relative system normalizers (Hall (5)) of a finite soluble group G. We choose an arbitrary normal subgroup X(p) of G for each prime p dividing the order of G, forming a normal system = {X(p)} of G. Each of these normal systems of G yields a conjugacy class of subgroups, called the -normalizers of G, which possess the above properties of -normalizers. However, they do not satisfy all the properties of the -normalizers unless is a so-called integrated normal system of G.

The definition and main result. It has been shown ((1), § 3) that if G is any finite group and p any prime number not dividing |G|, then the number of conjugacy classes of maximal nilpotent subgroups in the regular wreath product of a cyclic group of order p by G is equal to the number of conjugacy classes of all nilpotent subgroups in G. This fact, together with various properties of the map by means of which it was established, proved helpful in dealing with questions of construction raised in (1). The present note isolates the key property of the wreath product on which the argument rests, and from this shows how the argument can be carried over to a more general context. The essential situation is that a group G acts on a group A in a way which will be called ‘absolutely faithful’.

According to well-known theorems of Kaplansky and Baer–Kulikov–Kapla nsky–Fuchs (4, 2), the class of direct sums of countable Abelian groups and the class of direct sums of torsion-free Abelian groups of rank 1 are both closed under the formation of direct summands. In this note I give an example to show that the class of direct sums of torsion-free Abelian groups of finite rank does not share this closure property: more precisely, there exists a torsion-free Abelian group G which can be written both as a direct sum G = A⊕B of 2 indecomposable groups A, B of rank ℵ0 and as a direct sum G = ⊕n ε zCn of ℵ0 indecomposable groups Cn (nεZ) of rank 2, where Z is the set of all integers.

In a previous paper it was shown that a necessary and sufficient condition for two regular polynomial matrices T and U to have relatively prime determinants is that a certain matrix R be non-singular. This generalization of the resultant of two scalar polynomials is extended to give an expression for the sum of the degrees of the greatest common divisors of all pairs of invariant factors of T and U in terms of the rank of R. Further, if rank R exceeds a given value then specified pairs of factors are relatively prime.

The technique of sunny collapsing was introduced by Zeeman in (1) in order to prove the unknotting of PL ball pairs with co-dimension ≥ 3. Since then the technique has been used a number of times. In each case the method was used to prove special cases of the Theorem in this paper. Apart from providing a more unified treatment of various applications of this one technique, the theorem has important applications to the theory of relative regular neighbourhoods.

The purpose of this paper is to obtain some necessary conditions for a link in Euclidean 3-space to be spanned by a locally unknotted surface of given type in one half of 4-space. In particular necessary conditions for two links to be cobordant are proved.

A monothetic semigroup is a topological semigroup with jointly continuous multiplication which contains a dense cyclic subsemigroup. These semi-groups arise in a natural way in the study of semi-algebras. In (4) we showed that a compact monothetic semigroup in a Banach algebra can be characterized in terms of the spectral properties of a generating element. In this paper these spectral theorems are linked with the well-known structure theory of compact semigroups.

From results of Baker (2) it appeared to be unlikely that, in Hausdorff topological Abelian groups, completeness would be implied by Br- or B-completeness (for definitions, see below). We show here (Corollary 1) that the group of all complex roots of unity, though not complete, is B-complete. Another example, for the suggestion of which we are indebted to Dr J. W. Baker, is used to show (Corollary 2) that B-completeness is not a consequence of Br-completeness. It is proved, however, that B- and Br-complete Hausdorff topological Abelian groups are embedded in their completions in special ways in relation to the closed subgroups of the completions. Also, the completions of B- and Br-complete Hausdorff topological Abelian groups are shown to be, respectively, B- and Br-complete.

We extend the index theory of Fredholm operators in Banach spaces to general operators in normed linear spaces. In so doing we make use of a doubling technique, which we also apply to obtain improved results for the perturbation of semigroups and sesquilinear forms.

Our principal result deals with a function φ defined on the ‘non-negative half’ X+ of a discrete ordered group X, the dual of a compact Abelian connected group G. Suppose φ vanishes outside a countable set S = {χ1, χ2 …} and has the following property: for every t in a set 0 ⊂ (0, 1) of positive Lebesgue measure, the function on X+ defined to be 0 off S and to be rn(t)φ(χn) for χ = χn, where rn is the nth Rademacher function, is the restriction to X+ of a Fourier–Stieltjes transform. We conclude that φ ∈ ℓ2(X+).

Suppose that p > 1 and let Γ1, Γ2, …, Γp be Jordan arcs going from 0 to ∞ and having no other common point and numbered in anti-clockwise order around the origin. Let Γp+1 = Γ1 and let Dv be the domain bounded by Γv and Γv+1 for v = 1 to p.

Let f(z) be regular and univalent in |z| < 1 with a power series expansion

Then it is proved in (2) that

where z1 and z2 are points in |z| < 1.

In 1935 Tartakowski (7) proved that, in general, a cubic form in sufficiently many variables with coefficients in an algebraic number field K has a non-trivial zero in that field; and in the case when K is the rational field 57 variables suffice. Here, ‘in general’ means that the coefficients of the form do not lie in a proper subvariety of the coefficient space. Hence, Tartakowski's result holds for almost all cubic forms. Later, Lewis (5) proved that if K is any algebraic number field such that [K: Q] = n, then there exists a function ψ(n) such that every cubic form over K in m ≥ ψ(n) variables has a non-trivial zero in K. His bound, ψ(n), is extremely large; e.g. when K is the rational field, ψ(1) > 500.

Let GF(q) be the finite field containing q = pl elements, where p is a prime and l a positive integer. Let P(x) be a monic polynomial in GF[q, x] of degree m. In this paper we investigate the nature and distribution of monic irreducible polynomials of the following types:

(I) P(xr), where r is a positive integer (r-polynomials).

(II) xm P(x + x−1). (Reciprocal polynomials.) These have the form

(III) xrmP(xr + x−r). (r-reciprocal polynomials.) These have the form Q(xr), where q(x) satisfies (1·1).

In an earner paper ((6), p. 209) the author has given a generating function for the Jacobi polynomials defined by ((5), p. 254)

In the present note we obtain another generating function for these polynomials. The results given earlier by Manocha and Sharma ((4), p. 432) in these proceedings and Brafman in (2) follow as a particular case of our result.

In this paper we have employed Gauss's hypergeometric function, Legendre function and Jacobi polynomial to obtain the solutions of a fundamental differential equation of certain types of forced symmetrical vibrations of a circular elastic plate of finite radius.

Let be a given infinite series with the sequence of partial sums {sn}. Then the sequence-to-sequence Hausdorff transformation of the sequence {sn} is given by

In a recent paper Al-Salam(1) has denned a fractional q-integral operator by the basic integral

(1) Where α ≠ 0, −1, −2, …. Using the series definition of the basic integrals, (1·1) is written as

valid for all α

We denote by b(n) the number of ways of expressing the positive integer n as the sum of powers of 2 and we call b(n) ‘the binary partition function’. This function has been studied by Euler (1), Tanturri (2–4), Mahler (5), de Bruijn(6) and Pennington (7). Euler and Tanturri were primarily concerned with deriving formulae for the precise calculation of b(n), whereas Mahler deduced an asymptotic formula for log b(n) from his analysis of the functions satisfying a certain class of functional equations. De Bruijn and Pennington extended Mahler's work and obtained more precise results.

New methods are presented for the evaluation of lattice Green's functions for the body-centred cubic lattice.