Let denote the set of all normalized analytic univalent functions in the open unit disc D. Let f(z), F(z) and φ(z) be analytic in |z| < r. We say that f(z) is majorized by F(z) in we say that f(z) is subordinate to F(z) in where .

Let be the set of all locally univalent (f’(z) ≠ 0) analytic functions in D with order ≦α which are of the form f(z) = z +… . The family is known as the universal linear invariant family of order α [6]. A concise summary of and introduction to properties of linear invariant families which relate to the following material is contained in [1]. The present paper contains the proofs of some of the results announced in [1]

For 1 ≦ p < ∞, μ real, let Lμ, p denote the collection of functions f, Lebesgue measurable on (0, ∞ ), and such that ‖ f ‖μP < ∞ , where

(1.1)

Also, if X and Y are Banach spaces, denote by [X, Y] the collection of bounded linear operators from X to Y; [X, X] denote by [X].

Various characteristic conjugacy classes of subgroups having covering/avoidance properties with respect to chief factors have recently played a major role in the study of finite soluble groups. Apart from the subgroups which are now called Hall subgroups, P. Hall [7] also considered the system normalizers of a finite soluble group and showed that these form a characteristic conjugacy class, cover the central chief factors and avoid the rest. The system normalizers were later shown by Carter and Hawkes [1] to be the simplest example of a wealth of characteristic conjugacy classes of subgroups of finite soluble groups which arise naturally as a consequence of the theory of formations.

In this paper we derive a functional equation whose solution would give the sum of the chromatic polynomial P(M, λ) over certain classes of rooted planar maps M called “ triangulations” and “near-triangulations”. For an integral colour-number λ this sum is the number of λ-coloured rooted maps of the kind considered, but the sum can also be discussed for non-integral λ.

In an earlier paper [2] we denned the chromatic sums g, q, l and h. We determined the derivatives of these sums with respect to the colour-number λ at the special values λ = 1 and λ = 2. In the present paper we find parametric equations for h, l and q in the case λ = τ + 1, where τ is the golden ratio. We obtain h, l and the basic indeterminate z explicitly in terms of the parameter u, but for q we exhibit only a cubic equation with coefficients depending on u. We obtain an exact formula for the coefficients in h by applying Lagrange's theorem to the parametric equations.

Let K be a finite normal extension of an algebraic number field k; let k2 be the compositum of all quadratic extensions of k which are contained in K. Let ζk(s), ζK(s) and ζk2(s) denote the Dedekind ζ-functions of these fields.

In this paper we give sufficient conditions for an HNN group to have the following two properties:

(1) any two finitely generated subgroups intersect in a finitely generated subgroup;

(2) every finitely generated subgroup containing a non-trivial subnormal subgroup has finite index.

The following result is a particular case of the main theorem.

In this paper we shall study the sufficiency of complex jets. Let A (Cn, C) be the set of all analytic functions f : Cn → C with f (0) = 0. We call two functions f and g of A (Cn, C) equivalent of order r at 0 if, at 0, their Taylor expansions up to and including the terms of degree ≦ r are identical.

This paper continues the analysis of Clifford theory for the case of a finite group G, K a normal subgroup of G and G/K abelian which was developed in [7]. In [7] the permutation actions of G/K on the characters of K and of (G/K)^ on the characters of G were studied in relation to their effects on induction and restriction of group characters.

In [8] I. N. Herstein conjectured that all the finite odd order sub-groups of the multiplicative group in a division ring are cyclic. This conjecture was proved false in general by S. A. Amitsur in [1]. In his paper Amitsur classifies all finite groups which can appear as a multiplicative subgroup of a division ring. Let D be a division ring with prime field k and let G be a finite group isomorphic to a multiplicative subgroup of D.

Let Γ be the Fuchsian group with canonical presentation:

Generators:

Relations:

where . The genus g, the period partition {m1, m2, … , mr} and the number s are invariants of Γ. The symbol (g|m1, m2, … , mr|s) is called the signature of Γ and it determines Γ up to isomorphism.

A recent paper by Goro Azumaya on M-projective and M-injective modules [1] suggests a generalization of the concept of hereditary rings to modules which is also capable of dualization. Section 2 is devoted to preliminaries on M-projective and M-infective modules.

In section 3, we introduce the notion of hereditary and cohereditary modules. An R-module is called hereditary if every R-submodule of it is projective. Cohereditary modules are defined dually.