Let G be a p-primary Abelian group. Recall that the final rank of G is infn∈ω{r(pnG)}, where r(pnG) is the rank of pnG and ω is the first limit ordinal. Alternately, if Γ is the set of all basic subgroups of G, we may define the final rank of G by supB∈Γ {r(G/B)}. In fact, it is known that there exists a basic subgroup B of G such that r(G/B) is equal to the final rank of G. Since the final rank of G is equal to the final rank of a high subgroup of G plus the rank of pωG, one could obtain the same information if the definition of final rank were restricted to the class of p-primary Abelian groups of length ω.

Definition 1.1. Let be analytic for |z| < 1. If ƒ is univalent, we say that ƒ belongs to the class S.

Definition 1.2. Let ƒ ∈ S, 0 ≦ α < 1. Then ƒ belongs to the class of convex functions of order α, denoted by Kα, provided

(1)

and if > 0 is given, there exists Z0, |Z0| < 1, such that

Let ƒ ∈ Kα and consider the Jordan curve ϒτ = ƒ(|z| = r), 0 < r < 1. Let s(r, θ) measure the arc length along ϒτ; and let ϕ(r, θ) measure the angle (in the anti-clockwise sense) that the tangent line to ϒτ at ƒ(reiθ) makes with the positive real axis.

In his fundamental paper [3], Hurwitz showed that the order of a group of biholomorphic transformations of a compact Riemann surface S into itself is bounded above by 84(g – 1) when S has genus g ≧ 2. This bound on the group of automorphisms (as we shall call the biholomorphic self-transformations) is attained for Klein's quartic curve of genus 3 [4] and, from this, Macbeath [7] deduced that the Hurwitz bound is attained for infinitely many values of g.

After genus 3, the next smallest genus for which the bound is attained is the case g = 7. The equations of such a curve of genus 7 were determined by Macbeath [8] who also gave the equations of the transformations. The equations of these transformations were found by using the Lefschetz fixed point formula. If the number of fixed points of each element of a group of automorphisms is known, then the Lefschetz fixed point formula may be applied to deduce the character of the representation given by the group acting on the first homology group of the surface.

The approximately continuous integral which includes the Lebesgue integral has been considered by Burkill [1] and Ridder [3; 4]. I [2] also defined the AD-integral of this kind which is more general than the AP-integral of Burkill [1] and the Denjoy integral in the wide sense. But this integral is equivalent to the β-integral of Ridder.

Our aim in this paper is to characterize the AD-integral in the following way: The AD-integral is the least general approximately continuous integral (Definition 1) which includes the Lebesgue integral and fulfils the Cauchy and Harnack conditions (Definition 2).

The set operations under consideration are Borel operations and Souslin's operation (). With respect to a given family of sets and in a setting free of any topological structure there are defined three Borel families (Definitions 3.1) and the family of Souslin sets (Definition 4.1). Conditions on an initial family are determined under which iteration of the Borel operations with Souslin's operation () on the initial family and the families successively produced results in a non-decreasing sequence of families of analytic sets (Theorem 5.2.1 and Definition 3.5). A classification of families of analytic sets with respect to an initial family of sets is indicated in a manner analogous to the familiar classification of Borel sets (Definition 5.3).

1. The Frattini and Fitting subgroups of a finite group G have been useful subgroups in establishing necessary and sufficient conditions for G to be solvable. In [1, pp. 657-658, Theorem 1], Baer used these subgroups to establish several very interesting equivalent conditions for G to be solvable. One of Baer's conditions is that ϕ(S), the Frattini subgroup of S, is a proper subgroup of F(S), the Fitting subgroup of S, for each subgroup S ≠ 1 of G. Using the Fitting subgroup and generalized Frattini subgroups of certain subgroups of G we provide certain equivalent conditions for G to be a solvable group. One such condition is that F(S) is not a generalized Frattini subgroup of S for each subgroup S ≠ 1 of G. Our results are given in Theorem 1.

Let be a family of sets in a linear space X. A hyperplane π is called a k-secant of if π intersects exactly k members of . The existence of k-secants for families of compact sets in linear topological spaces has been discussed in a number of recent papers (cf. [3–7]). For X normed (and a finite family of two or more disjoint non-empty compact sets) it was proved [5] that if the union of all members of is an infinite set which is not contained in any straight line of X, then has a 2-secant. This result and related ones concerning intersections of members of by straight lines have since been extended in [4] to the more general setting of a Hausdorff locally convex space.

Let T be a linear transformation on Mn the set of all n × n matrices over the field of complex numbers, . Let A ∈ Mn have eigenvalues λ1, …, λn and let Er(A) denote the rth elementary symmetric function of the eigenvalues of A :

Equivalently, Er(A) is the sum of all the principal r × r subdeterminants of A. T is said to preserve Er if Er[T(A)] = Er(A) for all A ∈ Mn. Marcus and Purves [3, Theorem 3.1] showed that for r ≧ 4, if T preserves Er then T is essentially a similarity transformation; that is, either T: A → UAV for all A ∈ Mn or T: A → UAtV for all A ∈ Mn, where UV = eiθIn, rθ ≡ 0 (mod 2π).

Une structure de contact sur une variété M est un champ d'éléments de contact tangents Σ dont le système caractéristique Δ est régulier. Lorsque codim Σ = 1 une telle structure est définie localement par une forme de Pfaff ω ou encore par l'équation ω = 0. Le théorème classique de Darboux nous donne un critère d'équivalence de deux Pfaffiens ω et ω′ à l'aide de l'invariant classe ω. Cet invariant est cependant trop fort pour déterminer l'équivalence des structures de contact Σ et Σ′ définies par ω = 0 et ω′ = 0 respectivement. D'une façon précise, les structures Σ et Σ′ peuvent être équivalentes sans que les Pfaffiens ω et ω′ le soient. Dans cette note nous considérons un invariant plus faible, classe Σ, attaché à une structure de contact Σ et nous montrons qu'il détermine l'équivalence en codimension 1 (cf. corollaire 3).

Interest in the Singer groups has arisen in various places. The name itself results from the connection Singer [7] made between these groups and perfect difference sets, and this is closely associated with the geometric property that a Singer group is regular on the points of a projective space. Some information about these groups appears in Huppert's book [3, p. 187]. Singer groups are frequently useful in constructing examples and counterexamples. Our aim in this paper is to make a systematic study of the Singer subgroups of the linear groups, with a particular view to analyzing the examples they provide of Frobenius regular groups. Frobenius regular groups are a class of permutation groups generalizing the Zassenhaus groups, and Keller [5] has shown recently that they provide a new characterization of A6 and M11.

Let G be a separable locally compact group (separable in the sense that the topology of G has a countable base). Let S be a standard Borel space on which G acts on the right such that:

(1) s · g1g2 = (s · g1) · g2;

(2) s · e = s;

(3) (s, g) → s · g is a Borel function from S × G to S.

If μ is a Borel measure on S, let μg be the Borel measure on S defined by μg(E) = μ(E · g).

Let μ be a Borel measure on S which is quasi-invariant under the action of G; i.e., μg and μ are absolutely continuous (g ∈ G). The triple (G, S, μ) is called a dynamical system [11; 8].

Consider the following general problem. Let (G, S, μ) be a dynamical system.

Let cat be the Lusternik-Schnirelmann category structure as defined by Whitehead [6] and let be the category structure as defined by Ganea [2],

We prove that

and

It is known that w ∑ cat X = conil X for connected X. Dually, if X is simply connected,

1. We work in the category of based topological spaces with the based homotopy type of CW-complexes and based homotopy classes of maps. We do not distinguish between a map and its homotopy class. Constant maps are denoted by 0 and identity maps by 1.

We recall some notions from Peterson's theory of structures [5; 1] which unify the definitions of the numerical homotopy invariants akin to the Lusternik-Schnirelmann category.

In [2] Sandler studied generalizations of projective planes called pseudo planes. These structures gave rise to ternary operations for which addition and multiplication are loop operations. Our aim in this paper is to investigate the pseudo planes for which the operations of addition and multiplication give rise to nearfields. The pseudo planes for which this is true will be called π-systems in this paper, and a list of postulates for such systems is given in § 2. In that section some results on collineations of π-systems are given which are stronger than analogous results for general pseudo planes.

Let denote the class of all (fully) ordered groups satisfying the maximal condition on subgroups, and let denote the class of all locally groups. In this paper we investigate the family of convex subgroups of groups.

It is well known (see [1, pp. 51, 54]) that every convex subgroup of an is normal in G, and for any jump D –< C in the family of convex subgroups, [G′, C] ⊆ D. We observe that these properties are also true for any group and record, without proof, the following.

THEOREM 1. Any convex subgroup of angroup G is normal in G, and for any jump D –< C in the family of convex subgroups, [G′, C] ⊆ D.

1. Recently a number of functions have been shown to satisfy relations on polytopes similar to the classic Euler relation. Much of this work has been done by Shephard, and an excellent summary of results of this type may be found in [11]. For such functions, only continuity (with respect to the Hausdorff metric) is required to assure that it is a valuation, and the relationship between these two concepts was explored in [8]. It is our aim in this paper to extend the results obtained there to illustrate the relationship between valuations and the Euler relation on cell complexes.

To fix our notions, we will suppose that everything takes place in a given finite-dimensional Euclidean space X.

A polytope is the convex hull of a finite set of points and will be referred to as a d-polytope if it has dimension d. Polytopes have faces of all dimensions from 0 to d – 1 and each of these is in turn a polytope. A k-dimensional face will be termed simply a k-face.

Our study in this paper is two-fold: One is that of a semigroup of linear operators on the space C(S) of bounded continuous functions on a locally compact Hausdorff space S, while the other is that of a transition function of measures in the Banach space M(S) of bounded regular Borel measures on S. It will be seen that an informative and essentially non-restrictive theory of the former can be obtained when C(S) is given the strict topology rather than the usual supremum norm topology and that, in this setting, the natural relationship between semigroups and transition functions obtained when S is compact is maintained, essentially because the dual of C(S) with the strict topology is M(S).

Suppose that R is a commutative ring and G is a finite abelian group. In § 2 we review the definition of E(R, G) (T(R, G)), the group of all (commutative) Galois extensions S of R with Galois group G. We discuss the properties of these groups as functors of G and give an example which exhibits some of the pathological properties of the functor E(R, – ). In § 3 we display a homomorphism from E(R, G) to Pic (R(G)); we use this homomorphism to prove that if S is commutative, G has exponent m, and R(G) has Serre dimension 0 or 1, then a direct sum of m copies of S is isomorphic as a G-module to a direct sum of m copies of R(G). (This result is related to [5, Theorem 4.2], where it is shown that if S is a free R-module and G is any finite group with n elements, then Sn is isomorphic to R(G)n as G-modules.) We also give some examples of Galois extensions without normal bases.

Our aim in this work is to show that the global theory of group representations (as presented in [4] for example) is actually and naturally a part of the theory of cardinal algebras. In this sense it is analogous to the work of Kaplansky [2] and Loomis [3] on operator rings. In his paper Loomis proposed an abstract scheme for representation theory; it appears, however, that the idea of abstracting the equivalence class of a representation is more suitable. It is the set of all these classes that forms the cardinal algebra which we study.

The connection between cardinal algebras and operator theory has been made explicit by Fillmore [1], where he worked out a dimension theory for a class of cardinal algebras satisfying certain conditions. In the next section we shall discuss the relation between the two systems; we could say that the main difference lies in our introducing such axioms as to make possible the study of type III cases.

To prove [2, Theorem 1], it is enough to remark that if x is a point or line of F4 which is fixed under H, then x is fixed under each subgroup of H. By Sylow's theorem, H has a subgroup H’ of order three. By [1, p. 420, Lemma 2.2], it follows that x is in the subplane of F4 generated by those elements of {A, B, C, D} which are fixed under H'. This subplane consists of a point only. Hence, x is in {A, B, C, D}, which is impossible. This argument can be used in many other cases.

In this paper we study algebraic closures for commutative semiprime rings. The main interest, however, is with rings which are regular in the sense of von Neumann. These play the same role with respect to semiprime rings as fields do with respect to integral domains. Two generally distinct notions are defined: “algebraic” and “weak-algebraic” extensions. Each has the transitivity property and yields a closure which is unique up to isomorphism and is “universal”. Both coincide in fields.

The extensions here called “algebraic” were studied independently by Enochs [5] and myself. Our results on these extensions proceed from a different point of view, and allow us to answer a question posed by Enochs. Furthermore, these results are required (and were developed) in order to obtain the weak-algebraic closure, which was the original closure sought. The motivation for the weak-algebraic extensions is found in the work of Shoda [14, p. 134, no. 1].