In this paper we shall consider finite groups satisfying the following hypothesis.

Hypothesis I. Let G be a finite group which admits an automorphism σ of primeorder p, (p, |G|) = 1. Assume the fixed point subgroup B = CG(σ) contains some Sylow 2-subgroup.

In 1946 Krasnoselskii proved that if every n + 1 points of a compact connected set S in a Euclidean space En can see at least one point of S via S then S is starshaped [1]. This result was expanded by Robkin in 1965 [2]. In this paper, we characterize starshaped sets in En by a local arcwise convexity property relative to a point p.

Let U, C denote the open unit disk and unit circumference, respectively and G(z, w) be the Green's function on U. We say v is the Green's potential of a mass distribution v on U if

Throughout this paper we assume that every ring has unity and all modules are unital right modules. A ring R is called a (right) q-ring if every right ideal of R is quasi-injective [4], In this paper we study a generalization of this concept. A ring R is called a (right) weak q-ring (in short, wq-ring) if every right ideal of R, not isomorphic to RR, is quasi-injective. A ring R is called a right pq-ring if every proper right ideal of R is quasi-injective. Any upper triangular 2 X 2 matrix ring over a division ring is a wq-ring, which is not a q-ring. In Section 1, some general properties of wq-rings are established and, in particular, it is shown in (1.8) that a semiprime wq-ring has zero singular ideal.

There has been a lot of progress made on the finite-dimensional representations of species. In [3] and [11] the finite-dimensional representations of tame species are classified and in [13] it is shown that if S is a species of finite type, then every representation of 5 is a direct sum of finite-dimensional ones. However, comparatively little is known about infinite-dimensional representations.

Let ƒ be a complex-valued function denned on a closed subset F of the finite complex plane C, and let {Zn} be a sequence on F without limit points. We wish to find an analytic function g which simultaneously approximates ƒ uniformly on F and interpolates ƒ at the points {Zn}.

One aspect of topological analysis that authors, such as G. T. Whyburn and Marston Morse, have pointed to ([16; 6] for instance) as being fundamental in the development of function theory is the topological study of the level sets of analytic and harmonic functions or of their topological analogues, light open maps and pseudo-harmonic functions. The first step in this direction seems to have been made by H. Whitney [14] when he studied families of curves, given abstractly using a condition of regularity.

Throughout this paper the term ring will denote a commutative ring with unity and the term integral domain will denote a ring having no nonzero divisors of zero. The set of all prime ideals of a ring R can be viewed as a topological space, called the prime spectrum of R, and abbreviated Spec (R), where the topology used is the Zariski topology [1, Definition 4, § 4.3, p. 99]. The set of all prime ideals of R can also be viewed simply as aposet - that is, a partially ordered set - with respect to set inclusion. We will use the phrase the pospec of R, or just Pospec (/v), to refer to this partially ordered set.

The graphs are undirected, without loops or multiple edges. The edge set E(X) of a graph X is a set of certain unordered pairs [x, y] of distinct elements of the vertex set V(X). For x ϵ V(X) we denote by E(x; X) the edges of X incident with x. A (homo)morphism ϕ : X ⟶ Y is a function from V(X) to V(Y) which preserves edges; thus it induces ϕ# : E(X) ⟶ E(Y) by ϕ# [x, x’] = [ϕx, ϕx’].

In [1] Beurling and Helson prove the following theorem.

THEOREM. Let φ: R → R be a continuous map such that

Then φ is affine. (Here denotes the space of Fourier-Stieltjes transforms on R.)

By a polygon with n vertices (an n-gon) we shall mean an ordered n-tuple of complex numbers (z1, z2, … , zn). The numbers zi will be called the vertices of the polygon. The zi may or may not be distinct, and it shall not matter whether a “conventional” polygon can be drawn with the zi as vertices. We shall operate in the spirit of the book of Bachmann and Schmidt [2] though not at the same level of algebraic generality inasmuch as these authors allow zi to belong to a general field.

One of the main purposes of the present paper is to provide a proof for the following statement:

Theorem A. Let M be a complex manifold of a complex dimension n. Let be a fixed point in M such that there exists a square integrable holomorphic n-form a(z) on M with .

Let C2π denote the space of 2π-periodic continuous functions and πn the set of trigonometric polynomials of degree ≦ n, where n ϵ P = {0, 1, … } . Given θ > 0, the well-known theorem of Stečkin and its converse state that the best approximation of an ƒ ϵ C2π with respect to the max-norm satisfies

In [3] Hay proves generalizations of Rice's Theorem and the Rice-Shapiro Theorem for differences of recursively enumerable sets (d.r.e. sets). The original Rice Theorem [5, p. 3G4, Corollary B] says that the index set of a class C of r.e. sets is recursive if and only if C is empty or C contains all r.e. sets. The Rice-Shapiro Theorem conjectured by Rice [5] and proved independently by McNaughton, Shapiro, and Myhill [4] says that the index set of a class C of r.e. sets is r.e. if and only if C is empty or C consists of all r.e. sets which extend some element of a canonically enumerable class of finite sets. Since a d.r.e. set is a difference of r.e. sets, a d.r.e. set has an index associated with it, namely, the pair of indices of the r.e. sets of which it is the difference.

In [1] a simple but ingenious technique was developed for calculating hitting probabilities for submartingales (or martingales or supermartingales) subject to various constraints. This technique is extended here in order to find sharp bounds on upcrossing probabilities for submartingales subject to constraints.

Dirichlet's Theorem says that for any real α and for N ≧ 1, there exists a natural n ≦ N with

where || || denotes the distance to the nearest integer. Heilbronn [2], improving estimates of Vinogradov [3], showed that for α, N as above and for ϵ ≧ 0, there exists an n ≦ N with

1.1. Two complex measurable functions/ and g on complex measure spaces (X, η) and (Y, v) are equimeasurable, abbreviated ƒ ∼ g, if

for every Borel set E ⊆ C. If Φ is a continuous complex function on C, then we make the following standing hypothesis (HI) which relates Φ, f, and g:

(HI) For all α, β ∊ C, we have

In this paper we shall consider certain rank 3 permutation groups G which act on a set Ω of size n. Thus a point stabiliser Gα will have 3 orbits { α }, △ (α), Γ (α) of sizes 1, k, I respectively. It is well known that, if |G| is even, then the orbital △ defines a strongly regular graph on Ω. In this graph, every point has valency k, every pair of adjacent points are adjacent to a constant number λ of common points, and every pair of non-adjacent points are adjacent to a constant number μ of common points. This notation is reasonably standard (see [4], where much background theory is given).

Generalizing a celebrated theorem of Thompson, R. P. Martineau has established [4; 5] that a finite group which admits an elementary abelian group of automorphisms with trivial fixed-point subgroup is necessarily solvable. A critical observation in his approach to this problem is the fact that, corresponding to each prime divisor of its order, such a group contains a unique Sylow subgroup invariant (as a set) under the action. Hence, the theorem we shall derive here represents a modest extension of Martineau's result.

A Banach space X is called primary (respectively, prime) if for every (bounded linear) projection P on X either PX or (I — P)X (respectively, PX with dim PX = ∞ ) is isomorphic to X. It is well-known that C0 and lp, 1 ≦ p ≦ ∞ [8; 14] are prime. However, it is unknown whether there are other prime Banach spaces. For a discussion on prime and primary Banach spaces, we refer the reader to [9].