Our aim is to present some global results about the set of maximal abelian subgroups of the symmetric group Sn. We shall show that certain properties are true for “almost all” subgroups of this set in the sense that the proportion of subgroups which have these properties tends to 1 as n → ∞. In this context we consider the order and the number of orbits of a maximal abelian subgroup and the number of generators which the group requires.

Earlier results of this kind may be found in the papers [1; 2; 3; 4; 5]; the papers of Erdös and Turán deal with properties of the set of elements of Sn. The present work arose out of a conversation with Professor Turán who posed the general problem: given a specific class of subgroups (e.g., the abelian subgroups or the solvable subgroups) of Sn, what kind of properties hold for almost all subgroups of the class?

HNN groups have appeared in several papers, e.g., [3; 4; 5; 6; 8]. In this paper we use the results in [6] to obtain a structure theorem for the subgroups of an HNN group and give several applications.

We shall use the terminology and notation of [6]. In particular, if K is a group and {φi} is a collection of isomorphisms of subgroups {Li} into K, then we call the group

1

the HNN group with base K, associated subgroups { Li,φi(Li)} and free part the group generated by t1, t2, …. (We usually denote φi(Li) by Mi or L–i.) The notion of a tree product as defined in [6] will also be needed.

A space is realcompact if it is a homeomorph of a closed subspace of a product of real lines. Many external characterizations of realcompactness have appeared, but there seems to be no simple internal characterization. We provide such a characterization in terms of the existence of a collection of covers of a certain type and use it to examine realcompact extensions of a space and to characterize the Q-closure of a space in a compac tification.

A structure on X is a collection of covers of X that forms a filter under refinement ordering; the members of a structure are called gauges. A balanced refinement of a gauge a is a gauge β with cardinal not greater than that of a such that for each B ∈ β there is A ∈ α such that {A, X – B} is also a gauge; thus a balanced refinement is certainly a refinement.

The class of finite-to-one open mappings on manifolds contains some important subclasses. Any non-constant analytic function from a bounded region in its domain of definition is finite-to-one. Church [2] showed that any light strongly open Cn map f: Rn → Rn is discrete. A number of papers concerning discrete open mappings on manifolds have been published; see [1-6; 8-9; 11-14].

A result of Černavskiĭ [1] (see also [13]) shows that for any discrete strongly open mapping f : Mn → Nn of an n-manifold into an n-manifold, the branch set of f has dimension less than n – 1. If f is also a closed map, then N(f) is finite and the set of points x for which N(x, f) = N(f) is an open dense connected subset of Mn.

In this note we give a correction to the proof of the following theorem [1, Theorem 2].

THEOREM. Letbe a flexible, power-associative algebra, over an arbitrary algebraically closed field Ω of characteristic 0. Ifis a simple Lie algebra, thenis a simple Lie algebra isomorphic to.

Step (i) of the proof, which proves that the Cartan subalgebra of is a nil subalgebra of , is incomplete. Assuming that is not a nil subalgebra of , there exists an idempotent e ≠ 0 in .

Disconjugacy criteria have been established for linear selfadjoint differential equations of order 2n by Sternberg [4] and Ahlbrandt [1]. Such differential equations can be written in the form

1.1

where it is assumed that the coefficients are real and that Pn(x) ≠ 0. We shall be interested in nontrivial solutions v(x) of (1.1), which satisfy

1.2

for distinct points α and β. The smallest β> α such that (1.2) is satisfied nontrivially by a solution of (1.1), is denoted by μ1(α) and called the first conjugate point of x = α with respect to (1.1). If no such conjugate point exists we write μ1(α) = ∞, and say that (1.1) is disconjugate on [α, ∞).

The principal purpose of this paper is to generalize these disconjugacy criteria to the general linear nonselfadjoint differential equation of the form

1.3

A unitary polarity of a finite projective plane of order q2 is a polarity θ having q3 + 1 absolute points and such that each nonabsolute line contains precisely q + 1 absolute points. Let G(θ) be the group of collineations of centralizing θ. In [15] and [16], A. Hoffer considered restrictions on G(θ) which force to be desarguesian. The present paper is a continuation of Hoffer's work. The following are our main results.

THEOREM I. Let θ be a unitary polarity of a finite projective planeof order q2. Suppose that Γ is a subgroup of G(θ) transitive on the pairs x, X, with x an absolute point and X a nonabsolute line containing x. Thenis desarguesian and Γ contains PSU(3, q).

Dans tout ce qui suit, R désigne un anneau unitaire. Soit M R un R-module à droite; M R est muni canoniquement d'une structure de module à gauche sur l'anneau Λ de ses endomorphismes, pour laquelle il sera noté AM.

On sait [3, Théorème 3.1] que si M R est quasi-injectif à sous-module singulier nul, Λ est un anneau régulier au sens de Von Neumann, auto-injectif à droite. Dans [5] B. Osofsky étudie l'auto-injectivité à gauche de Λ et pose les problèmes suivants.

(1) Si MR est un module injectif à sous-module singulier nul, et si Λ est auto-injectif à gauche, ΛM est-il injectif?

(2) Si Ton remplace l'hypothèse M R injectif par M R quasi-injectif, ΛM est-il quasi-injectif?

On apporte ici une réponse affirmative à ces deux questions lorsque R est un anneau commutatif (Théorèmes 7 et 8).

C-matrices appear in the literature at various places; for a survey, see [11]. Important for the construction of Hadamard matrices are the symmetric C-matrices, of order v ≡ 2 (mod 4), and the skew C-matrices, of order v ≡ 0 (mod 4). In § 2 of the present paper it is shown that there are essentially no other C-matrices. A more general class of matrices with zero diagonal is investigated, which contains the C-matrices and the matrices of (v, k, λ)-systems on k and k + 1 in the sense of Bridges and Ryser [6]. Skew C-matrices are interpreted in § 3 as the adjacency matrices of a special class of tournaments, which we call strong tournaments. They generalize the tournaments introduced by Szekeres [24] and by Reid and Brown [21].

0. In [3], Fell introduced a topology on Rep (A,H), the collection of all non-null but possibly degenerate *-representations of the C*-algebra A on the Hilbert space H. This topology, which we will call the Fell topology, can be described by giving, as basic open neighbourhoods of π0 ∈ Rep(A, H), sets of the form

where the ai ∈ A, and the ξj ∈ H(π0), the essential space of π0 [4].

A principal result of [3, Theorem 3.1] is that if the Hilbert dimension of H is large enough to admit all irreducible representations of A, then the quotient space Irr(A, H)/∼ can be identified with the spectrum (or “dual“) Â of A, in its hull-kernel topology.

This paper sets forth conditions sufficient that the generalized Lototsky method (F, dn) be regular and Mercerian. If the dn's are real and of constant sign, then the conditions are also necessary. Moreover, it follows that if f is a polynomial, then under the same conditions the method (f,dn) is equivalent to the Sonnenschein method generated by f . Various related results are also given.

Definition 2.1. Let f be a nonconstant function holomorphic on the closed unit disk and let be a complex sequence with f (l) + dn ≠ 0. Suppose

1. Let A denote a sequence to sequence transformation given by the normal matrix A = (ank)(n, k = 0, 1, 2, …), i.e., a lower triangular matrix with ann ≠ 0 for all n. For B = (bnk) we write B ⇒ A if every B limitable sequence is A limitable to the same limit, and say that B is equivalent to A if B ⇒ A and A ⇒ B. If B is normal, then it is well known that the inverse of B exists (we denote it by B-l) and that B ⇒ A if and only if F = AB-1 is a regular transformation, i.e., transforms every convergent sequence into a sequence converging to the same limit. We say that a series ∑ an† is summable A if its sequence of partial sums is A-limitable.

Let k be a local field; that is, a complete discrete-valued field having a perfect residue class field. If L is a finite Galois extension of k then L is also a local field. Let G denote the Galois group GL|k. Then the nth ramification group Gn is defined by

where OL, denotes the ring of integers of L, and PL is the prime ideal of OL. The ramification groups form a descending chain of invariant subgroups of G:

1

In this paper, an attempt is made to characterize (in terms of the arithmetic of k) the ramification filters (1) obtained from abelian extensions L\k.

In [3] Fox studied a certain class of irregular coverings of S3 branched along some knot or link which turned out to be homotopy spheres. By a simple geometric construction, it is shown in this paper that these homotopy spheres are just 3-spheres, provided that the group of the knot or link k in question cannot be generated by a number of Wirtinger generators smaller than the minimal number of bridges of this knot or link. The knots and links with two bridges provide examples for such coverings. In the covering sphere there is a link covering k. With the help of braid automorphisms, can be determined. Figure 3 shows a link in a 5-sheeted covering over k = 41. Links over 31 and 61 in 3-sheeted coverings were determined by Kinoshita [5] by a different method. The method used here is applicable to these cases and confirms his results.

In 1883 Poincaré [13] recognized that the discrete subgroups of PSL(2, C) could be extended from their natural action on the complex plane to acting on hyperbolic 3-space and he attempted to analyze these groups in an analogous manner to his classical treatment of Fuchsian groups, with fundamental polyhedra playing the role of the fundamental polygons for Fuchsian groups. This approach, however, did not lead very far, perhaps not surprisingly when one appreciates the close connection between the geometry of these groups and the topology of 3-manifolds. Since that time the state of knowledge remained essentially unchanged until 1964 when work by Ahlfors [1] and soon afterwards by Bers [3] revitalized the subject of Kleinian groups. The modern approach tends to use analytic methods, although recently Marden [11] has had considerable success in carrying forward Poincaré's geometric approach.

In [5], Rinehart showed that if X is an n × n complex matrix with distinct eigenvalues, then a suitably defined diagonalizing matrix P and the diagonal matrix Λ of eigenvalues in P–lXP = Λ are both Hausdorff differentiate functions in an open set containing X. Furthermore, if the scalar function ƒ(z) is analytic at the eigenvalues of X, then the primary matrix function ƒ(X) is Hausdorff differentiable, and its differential may be represented in terms of the differentials of P and Λ [4]. Rinehart noted that the actual computation of differentials was difficult and ad hoc. This difficulty clearly arises because of the definition given for the diagonalizing matrix. Therefore, our aim in this note is to give a different definition of the diagonalizing matrix, one which simplifies the computations.

In this paper we study the problem of representing groups as groups of automorphisms on an orthomodular lattice or poset. This problem not only has intrinsic mathematical interest but, as we shall see, also has applications to other fields of mathematics and also physics. For example, in the “quantum logic” approach to an axiomatic quantum mechanics, important parts of the theory can not be developed any further until a fairly complete study of the representations of physical symmetry groups on orthomodular lattices is accomplished [1].

We will consider two main topics in this paper. The first is the analogue of Schur's lemma and its corollaries in this general setting and the second is a study of induced representations and systems of imprimitivity.

In this paper, we obtain information about the minimal degree δ of any non-trivial projective representation of the group PSL(n, q) with n ≧ 2 over an arbitrary given field K. Our main results for the groups PSL(n, q) (Theorems 4.2, 4.3, and 4.4) state that, apart from certain exceptional cases with small n, we have the following rather surprising situation: if q = pf (where p is a prime integer) and char K = p, then δ = n, but if q = pf and char K ≠ p, then δ is of a considerably higher order of magnitude, namely, δ is at least qn–l – 1 or if n = 2 and q is odd. Note that for n = 2, this lower bound for δ is the best possible. However, for n ≧ 3, this lower bound can conceivably be improved.

In the study of inequalities, the cases of equality are often the most difficult and interesting part. The case of equality is, in some sense, a measure of the tightness of the inequality. In this paper, we generalize two inequalities of Brualdi and Newman [1, Theorems 3, 4], but the instances of equality are probably more interesting because of the variety of cases which can occur.

Let A = (aij) be an n × n matrix. Define the permanent of A by

We say that A is row stochastic if all entries are non-negative and all row sums are 1. In [1], several inequalities involving permanents of row stochastic matrices were proved. In two of these results, the case of equality was not determined. We will generalize both of these results to a class of functions which includes the permanent, and determine all cases of equality.