An index four simple group is a finite simple group, G, with a self-centralizing Sylow p-subgroup whose normalizer in G has order 4p. In this paper index four simple groups having a non-principal ordinary irreducible character of small degree in the principal p-block are studied.

Furnishing the open unit ball of a complex Hilbert space with the Carathéodory-differential metric, we construct a model which plays a similar role as that of the Poincaré model for the hyperbolic geometry.

In this note we study the question whether or not through a point in the model not lying on a given line there exists a unique perpendicular, and give a necessary and sufficient condition for the existence of a unique perpendicular. This enables us to divide a triangle into two right triangles. Many trigonometric identities in a general triangle are easy consequences of various identities which hold on a right triangle.

There are theorems in which some classes of topological spaces are characterized by means of properties of mappings of these spaces into a single space. For example, it is well known that a compactum X is at most n-dimensional if and only if no mapping of X irto an (n + l)-cube has a stable value [5, Theorems VI. 1-2, pp. 75-77]. Also, a curve X is tree-like if and only if no mapping of X into a figure eight is homotopically essential [1, Theorem 1, pp. 74-75; 8, p. 91]. By a curve we mean any at most 1-dimensional continuum; a continuum is a connected compactum; a compactum is a compact metric space, and a mapping is a continuous function. The aim of the present paper is to prove another theorem of this type. We distinguish a class of curves and show that it is characterized by imposing the condition that no weakly confluent mapping [13] can transform the given curve onto a simple triod (see 2.4). A related result is applied to a generalized branch-point covering theorem (see 3.2). In addition, two results are obtained in which we establish some characterizations of weakly confluent images and preimages of the product of the Cantor set and an arc (see 1.1 and 2.2). Continua that are such images turn out to be identical with regular curves (see 1.3).

The Brown-Peterson spectrum BP has been used recently to establish some new information about the stable homotopy groups of spheres [9; 11]. The best results have been achieved by using the associated homology theory BP*( ), the Hopf algebra BP*(BP), and the Adams-Novikov spectral sequence

The theory of measure on topological spaces has in recent years found its most natural setting in the study of pavings and measures on such pavings (see e.g. [1-3; 5; 6; 10; 19; 22; 32; 33]. In this setting the relationship between measure and topology crystallizes since one concentrates primarily on the simpler internal lattice structure associated with sublattices of the topology rather than on the more complex topological structure.

A Banach algebra P over C with identity element is called an N-algebra if any closed ideal in P is the intersection of maximal ideals. An example is given by the algebra of the continuous C-valued functions on a compact Hausdorff space X under the supremum norm; two others are discussed in § 3.

The purpose of this paper is to consider, in rings with involution, the structure of those subrings which are invariant under Lie commutation with [K, K]. Our goal is to find conditions which force such subrings to contain a noncentral ideal of the ring. Of course, the subring itself may lie in the center. Orders in 4 × 4 matrix rings over fields are known to provide examples of invariant subrings which are not central and contain no ideal (see [1, 40] or [5]). Except for subdirect products of these two kinds of “counter-examples”, we show that in semi-prime rings, invariant subrings do contain noncentral ideals. This generalizes work of Herstein [4] in two directions by considering semi-prime rings rather than simple rings, and by using [K, K] instead of K.

All rings in this paper are assumed to be commutative with identity, and the undefined terminology is the same as that in [3].

In 1956, in an important paper [2], M. Nagata constructed an example which showed (among other things): (i) a maximal chain of prime ideals in an integral extension domain R' of a local domain (R, M) need not contract in R to a maximal chain of prime ideals; and, (ii) a prime ideal P in R' may be such that height P < height P ∩ R. In his example, Rf was the integral closure of R and had two maximal ideals. In this paper, by using Nagata's example, we show that there exists a finite local integral extension domain of D = R[X](M,X) for which (i) and (ii) hold (see (2.8.1) and (2.10)).

Let G1 and G2 be compact groups and T : Lp(G1) → LP(G2) (1 ≦ P ≦ ∞ ) be an algebra homomorphism. If || T || ≦ 1 and T is either a monomorphism of an epimorphism then T can in many cases be explicitly characterized (see [4 ; 8 ; 9 ; 11 ; 13 ; 14]). Excluding p = 2, the outstanding cases are 1 < p < ∞ for monomorphisms and 2 < p < ∞ for epimorphisms (cf. [14]). One aim of the present note is to complete this work. We also consider the problem of extending these results in some form to homomorphisms on ideals of group algebras; the only known result in this area is for abelian groups [3].

In 1964, J. J. Charatonik in [1] introduced a new class of mappings, the so-called confluent mappings, which comprises the classes of open, monotone and quasi-interior mappings (see [20]). In 1966, A. Lelek started working on confluent mappings with applications to continua theory (see [7]). He introduced two other classes of mappings, the so-called weakly confluent and pseudo confluent mappings, he proved the invariance of rational continua under open, monotone and quasi-interior mappings and he asked about their invariance under confluent mappings. In 1976, E. D. Tymchatyn gave an example of a confluent mapping, which does not preserve the rationality of a curve (see [18]).

In this paper we consider a generalization of the discrete Poisson kernel for the Hahn polynomials obtained recently by Gasper [6]. The Hahn polynomials of degree n are defined by

(1.1)

and are known to be orthogonal on the set of non-negative integers x = 0, 1, . . . , N provided Re α, β > - l or Re α, β < -N [7; 8].

This paper completes a study, begun in [7], of conditions under which a generalized Hankel conjugate transformation Hƛ is bounded between a pair of μα -rearrangement invariant function spaces, the measure μα being defined by dμα(t) = tα∼1dt. Examples of such spaces are the Lp(μα) of Lebesgue and generalizations of them due respectively to Orlicz and Lorentz.

L. L. Herrington recently characterized completely Hausdorff-closed topological spaces in terms of arbitrary filterbases and a type of convergence for filterbases called f-convergence [2], Before this, characterizations of these spaces via filterbases was by open filterbases [1]. In this article, we employ Herrington's characterizations to obtain characterizations of completely Hausdorff-closed spaces in terms of projections and in terms of graphs of functions into the spaces; both of these—projections and graphs—are utilized in conjunction with a class S of spaces containing as a subclass the Hausdorff completely normal fully normal spaces to effect the characterizations.

Let D be a division algebra and let D* denote the multiplicative group of nonzero elements of D. In [3] Herstein and Scott asked whether any subnormal subgroup of D* must be normal in D*. Our purpose here is to show that division algebras over certain p-local fields do not satisfy such a “subnormal property”.

Let K be a field complete for a proper valuation (absolute value) v. It is classic that a finite-dimensional K-vector space E admits a unique Hausdorff topology making it a topological K-vector space, and that that topology is the “cartesian product topology” in the sense that for any basis c1 …, cn of E, is a topological isomorphism from Kn to E [1, Chap. I, § 2, no. 3; 2, Chap. VI, § 5, no. 2]. It follows readily that any multilinear mapping from Em to a Hausdorff topological K-vector space is continuous. In particular, any multiplication on E making it a K-algebra is continuous in both variables. If for some such multiplication E is a field extension of K, then by valuation theory the unique Hausdorff topology of E is given by a valuation (absolute value) extending v.

Suppose throughout that c is a fixed positive integer, that

and that

In the recent paper [8] Phillips has shown that for each prime p there are 2ℵ0 non-isomorphic 2-generatecl p-groups. This same result was obtained independently by S. Jeanes and J. S, Wilson (unpublished) who show that the groups constructed in [1] have 2ℵ0 non-isomorphic images. The groups in both of these proofs all have infinite exponent. In this paper we show that, for large enough primes p, there are 2ℵ0 non-isomorphic 2-generated groups of exponent p2.

The real interpolation method is a very convenient tool in the study of imbedding relationships among Sobolev spaces and some of their fractional order generalizations, (Besov spaces, Nikolskii spaces etc.) Central to the application of these methods is the a priori determination that a given Sobolev space Wk'p(Ω) belongs to an appropriate class of spaces intermediate between two other “extreme” spaces.

In this paper we study subrings A of generated by monomials over . If A is normal and A ⊂ B integral, we can completely characterize A. If dim A = 2, we show that A is isomorphic to a subring A’ of B generated by monomials with A’ ⊂ B integral. The author became interested in these rings while studying projective modules over subrings of , For some applications, see [1].