In [6] Sands proved that the semisimple classes of associative rings are exactly the coinductive and closed under ideals and extensions classes. This characterization was transferred to the alternative case by Van Leeuwen, Roos and Wiegandt in [3]. Answering a question of [9], Sands [7] has recently proved that in the associative case the condition of being closed under ideals can be replaced by the regularity of the class. The same result for alternative rings has been proved by Anderson and Wiegandt in [2]. Thus the following result holds.

In what follows, character means irreducible complex character.

Let G be a finite group and let % be a character of a normal subgroup N. If χ extends to a character of G then χ is stabilised by G, but the converse is false. The aim of this paper is to prove the following theorem which gives a sufficient condition for χ to be extended to a character of G.

The problem of deciding which graded polynomial algebras over the field of p elements can occur as the -cohomology of a space has played a central rôle in the development of algebraic topology beginning as early as 1950. In the case where the polynomial generators do not occur in dimensions divisible by p, Adams and Wilkerson [1] have given a complete solution by showing that the spaces constructed by Clark and Ewing [3] suffice to realize all such algebras as -cohomology rings. The main result of Adams and Wilkerson for odd primes can be stated as follows.

By analogy with the concept of “inverse semi-group” in semi-group theory, in this paper we introduce the concept of “generalized near-field” in near-rings. A near-ring N is called a generalized near-field (GNF) if for each a ε N there exists a unique b ε N such that a = aba and b = bab, that is (N, ·) is an inverse semi-group. Surprisingly, this concept in rings coincides with that of “strong regularity”. But this is not true in the case of near-rings. Every GNF is strongly regular, but in general the converse is not true.

In [4] Elmer Rees proves that the symplectic group Sp(n) can be smoothly embedded in Euclidean space with codimension 3n, and the unitary group U(n) with codimension n. These are special cases of a result he obtains for a compact connected Lie group G. The general technique is first to embed G/T, where T is a maximal torus, as a maximal orbit of the adjoint representation of G, and then to extendto an embedding of G by using a maximal orbit of a faithful representation of G. In thisnote, we observe that in the cases G = Sp(n) or SU(n) an improved result is obtained byusing the “symplectic torus” S3 x … x S3 in place of T = S1 x … x S1. As in Rees's construction, the normal bundle of the embedding of G is trivial.

Let (S, Σ, μ) and (T, Θ, v) be two measure spaces of finite measure where we assume S, T are compact Hausdorff spaces and μ, v are regular Borel measures. We construct the product measure space (T x S, >, Φ σ) in the usual way. Let G = [gl, g2, …, gp] and H = [hl, h1, …, hm be finite dimensional subspaces of C(S) and C(T) respectivelywhere G and H are also Chebyshev with respect to the L1-norm. Note that a subspace Y of a normed linear space X is Chebyshev if each x ∈X possesses exactly one best approximation y ∈Y. For example, in C(S) with the L1-norm, the subspace of polynomials of degree at most n is a Chebyshev subspace. This is an old theorem of Jackson. Now set

If G is a finite group and A is a group of automorphisms of G, the “centralizer” nearring C(A, G) consists of the identity-preserving maps from G to itself which commute with the action of A. The main concern of this paper will be with the additive structur of C(A, G) in the case that this near-ring is semisimple.

Let G be a p-solvable group with a p-Sylow subgroup P of order pa and let t(G) be the nilpotency index of the radical of a group algebra of G over a field of characteristic p. The purpose of this paper is to give an elementary proof of the following result of Koshitani [1, Theorem].

In Theorems 1 and 2 of [] necessary and sufficient conditions were given for a group G to have a finite automorphism group Aut G and a semisimple subgroup of central automorphisms AutcG. Recently it occurred to us, as a result of conversations with Ursula Webb, that these conditions could be stated in a much simpler and clearer form. Our purpose here is to record this reformulation. For an explanation ofterminology and notation we refer the reader to [1].

According to Mason [1] a right near-ring N is called (i) left (right) strongly regular if for every a there is an x in N such that a = xa2 (a = a2x) and (ii) left (right) regular if for every a there is an x in N such that a = xa2 (a = a2x) and a = axa. He proved that for a zerosymmetric near-ring with identity, the notions of left regularity, right regularity and left strong regularity are equivalent. The aim of this note is to prove that these three notions are equivalent for arbitrary near-rings. We also show that if N satisfies dec on iV-subgroups, then all the above four notions are equivalent.

Let F be any field of characteristic p > 0, G a finite p-solvable group, pa the order of Sylow p-subgroups of G, FG the group algebra of G over F, and J(FG) the Jacobson radical of FG. Following Wallace [11] we write t(G) for the least integer t≧1 such that J(FG)t = 0.

Let G be a group, written additively with identity 0, but not necessarily abelian and let S be a semigroup of endomorphisms of G. The set for all is a zero-symmetric near-ring with identity under the operations of function addition and composition, called the centralizer near-ring determined by the pair (S, G). Centralizer near-rings are general, for if N is any zero-symmetric near-ring with identity then there exists a group G and a semigroup S⊇ G such that For background material and definitions relative to near-rings in general we refer the reader to the book by Pilz [7]. For material on centralizer near-rings we refer the reader to [4] and [6].

The paper is concerned with a curve F, the complete intersection of a quadric with a quartic surface, that admits a group of self-projectivities isomorphic to the symmetric group of degree 5. Every generator of the quadric is, as shown at the end of the paper, cut by F equianharmonically. F has 80 stalls, points where its osculating plane is stationary; they are of two kinds, 60 to be labelled ∑, the other 20 Ω. F also has inflections at 24 points which compose a figure encountered on earlier occasions. A search is made for tritangent planes of F of which, when reckoned according to proper multiplicity, there must be 2048. Among them are 60 all of whose three contacts are ∑ while a further 120 each involve a single ∑ among their contacts and 420 each involve a single Ω.