The maximum genus γM(G) of a connected graph G has been defined in [2] as the maximum g for which there exists an embedding h : G —> S(g), where S(g) is a compact orientable 2-manifold of genus g, such that each one of the connected components of S(g) — h(G) is homeomorphic to an open disk; such an embedding is called cellular. If G is cellularly embedded in S(g), having V vertices, E edges and F faces, then by Euler's formula

V-E + F = 2-2g.

Suppose Q is a q-group for a prime q and C ≦ Aut(Q) is cyclic of order pe for a prime p ≠ q. Let k be a splitting field for CQ, the semidirect product of C by Q, of characteristic r ≠ q. Let V be a faithful irreducible k[CQ]-module. The k[CQ]-module V has been widely studied. When r = p the situation outlined above is similar to the situation occurring in the proof of Theorem B of Hall and Higman [3]. When r ≠ p it is similar to the corresponding theorem of Shult [5]. In all but restricted cases V|C has a regular k[C]-direct summand.

It is known that the following classes of spaces (all spaces in this article are assumed T1) are identical:

1. Images of metric spaces under continuous open maps with compact point inverses.

2. Spaces with uniform bases (in the sense of Alexandrov [1]).

3. Metacompact developable spaces.

4. Spaces with σ-point-finite bases in which closed sets are Gδ's.

Let F denote a Galois field of order q and odd characteristic p, and F* = F\{0}. Let Sn denote an n-dimensional affine space with base field F. E. Cohen [1] had proved that if n ≧ 4, there is no hyperplane of Sn contained in the complement of the quadric Qn of Sn defined by

1.1

and in S3, there are q + 1 or 0 planes contained in the complement of Q3 according as — aα is not or is a square of F.

The use of computers to investigate groups has mainly been restricted to finite groups. In this work, a method is given for finding all subgroups of finite index in a given group, which works equally well for finite and for infinite groups. The basic object of study is the finite set of cosets. §2 reviews briefly the representation of a subgroup by permutations of its cosets, introduces the concept of normal coset numbering, due independently to M. Schaps and C. Sims, and describes a version of the Todd-Coxeter algorithm. §3 contains a version due to A. Dietze of a process which was communicated to J. Neubuser by C. Sims, as well as a proof that the process solves the problem stated in the title. A second such process, developed independently by M. Schaps, is described in §4. §5 gives a method for classifying the subgroups by conjugacy, and §6, a suggestion for generalization of the methods to permutation and matrix groups.

The geometries considered here are the Möbius plane M() (W. Benz [1]), the Laguerre plane L() (W. Benz and H. Mäurer [7]) and the Minkowski plane A() (W. Benz [5], G. Kaerlein [18]) over a field . All of them are geometries of an algebra with identity over a field.

The characterization of the projective plane over a field by the proposition of Pappus first gave a close relation between algebraic and geometric structures. B. L. v. d. Waedern and L. J. Smid [28] presented a further example by characterizing the Möbius and Laguerre plane with incidence axioms and the "complete" proposition of Miquel.

It is known that there are a lot of properties of the group of a knot in S3 which fail to generalize to the group of a knotted sphere in S4; among them are included Dehn's lemma, Hopf's conjecture, and the aspherity of knots. In this paper, we shall investigate the properties of the fundamental groups of regions in S3 and in S4, with examples to show that they are not quite the same. Some special consideration will be given to regions that are the complements in S3 or in S4 of a finite number of tamely imbedded manifolds of co-dimension 2, and, more generally, to regions that are the complements of subcomplexes in S3 or in S4.

If D is an integral domain with quotient field K, the group of divisibility G(D) of D is the partially ordered group of non-zero principal fractional ideals with aD ≦ bD if and only if aD contains bD. If K* denotes the multiplicative group of K and U(D) the group of units of D, then G(D) is order isomorphic to K*/U(D), where aU(D) ≦ bU(D) if and only if b/a ∊ D.

First we will briefly define the [f, g] and Aλ summability methods. Let K = {w : |w| < 1}. T. H. Gronwall [3] introduced a general class of summability methods each of which involves a pair of functions f and g with the following properties. The function z = f (w) is analytic on \{1}, continuous and univalent on , with f (0) = 0, f (1) = 1, |f(w)| < 1 if w ∊ K. The inverse function w = f-1(z) is analytic on f (K)\{1}, and at z = 1

1.1

where γ ≧ 1, a > 0, and the quantity in brackets is a power series in 1 — z with positive radius of convergence.

Let V denote a ring of subsets of an abstract space X, let R denote the real numbers, and let N denote the positive integers. Denote by a(V, R) (respectively ca(V, R)) the space of real valued, finitely additive (respectively countably additive) functions on the ring V and denote by ab(V, R) the subspace consisting of those members of the space a(V, R) with finite variation on each set in the ring V. Members of the space a(V, R) are referred to as charges and members of the space ab(V, R) are referred to as locally bounded charges. We denote by cab(V, R) the intersection of the spaces ab(V, R) and ca(V, R).

In [7] Richardson constructed a Stone-Čech type compactification R(E) of a Hausdorff convergence space E. Two questions arise in this regard. First, when is R(E) homeomorphic to β(E), β(E) the topological Stone-Čech compactification of E, for a Tychonoff topological space E? Second, if E is a regular convergence space, when is R(E) regular? The last question is motivated by the study of regular compactifications in [6]. In section 2 it will be shown that a necessary and sufficient condition in answer to both questions, is that α = cl(α) for each nonconvergent ultrafilter α on E.

In this paper all rings are commutative and have a unity. All ring homomorphisms preserve the unity. We let L denote the standard language for rings with two distinct constants, 0 and 1, playing the role of the zero and the unity respectively. A ring is regular if it satisfies the axiom (∀r) (∃r′)(rr′r = r) and it is algebraically closed if, for each integer n ≧ 1, it satisfies the sentence

Let be a fixed sequence of real or complex numbers. The Hausdorff transform {tn} of a sequence \sn) by means of the fixed sequence (or, in short, the (H, μn) transform) is given by

where, for r, q ≧ 0,

It is well-known that every weak basis in a Fréchet space is actually a basis. This result, called the weak basis theorem was first given for Banach spaces in 1932 by Banach [1, p. 238], and extended to Fréchet spaces by Bessaga and Petczynski [3]. McArthur [12] proved an analogue for bases of subspaces in Fréchet spaces, and recently W. J. Stiles [18, Corollary 4.5, p. 413] showed that the theorem fails in the non-locally convex spaces lp (0 < p < 1). The purpose of this paper is to prove the following generalization of Stiles' result.

Let V denote a ring of subsets of an abstract space X, let R+ denote the nonnegative reals, and let N denote the set of positive integers. We denote by C(V) the space of all subadditive and increasing functions, from the ring V into R+, which are zero at the empty set. The space C(V) is called the space of contents on the ring V and elements are referred to as contents.

A sequence of sets An ∊ V, n ∊ N is said to be dominated if there exists a set B ∊ V such that An ⊆ B, for n = 1, 2, A content p ∊ C(V) is said to be Rickart on the ring V if limnp(An) = 0 for each dominated, disjoint sequence An ∊ V, n ∊ N.

In this note we prove some results which assert that under certain conditions the involution on a prime ring must satisfy a form of positive definiteness. As a consequence of the first of our theorems we obtain a fairly short and simple proof of a recent theorem of Lanski [3]. In fact, in doing so we actually generalize his result in that we need not avoid the presence of 2-torsion. One can easily adapt Lanski's original proof, also, to cover the case in which 2-torsion is present. This result of Lanski has been greatly generalized in a joint work by Susan Montgomery and ourselves [2].

Let S be a semigroup. A function λ(ρ) on S is a left (right) translation of S if, for all x, y ∊ S, λ(xy) = λ(x)y ((xy)ρ= x(yρ)). A left translation λ and a right translation ρ are said to be linked if x(λy) = (xρ)y, for all x,y ∊ S,and then the ordered pair (λ, ρ) is called a bitranslation. Clearly the set Λ(S) (P(S)) of all left (right) translations is a semigroup with respect to composition of functions. The set of bitranslations forms a subsemigroup of the direct product Λ(S) × P(S)which is called the translational hull, Ω(S), of S. A valuable survey of results relating to Ω(S) and its importance in relation to semigroup extensions will be found in Petrich's review [6], to which the reader is referred for basic results on translational hulls.

In [1] Atiyah described how to use the complex representations of the symmetric group, Sn, to define and investigate operations in complex topological K-theory. In this paper operations for more general Grothendieck groups are described in terms of the integral representations of Sn using the representations directly without passing to the dual as Atiyah did. The principal tool, which is proved in the first section, is the theorem that the direct sum of the Grothendieck groups of finite integral representations of Sn form a bialgebra isomorphic to a polynomial ring with a sequence of divided powers. A consequence of this theorem is that the only operations that can be constructed from the symmetric groups will be polynomials in the symmetric powers.

It is well known that a real valued continuous function f on a closed interval S assumes every value between its maximum and minimum on S, i.e. if ξ is such that f(α) ≦ ξ ≦ f(β) then there exists γ between α and β such that f(γ) = ξ. The purpose of this paper is to develop the existence theory associated with differential and integral inequalities in the context of an intermediate value property for operators on partially ordered spaces. This has the advantage of allowing rather simple proofs of known results while in most cases giving slight improvements, and in some cases substantial improvements, in these results. Classical and recent results from different areas are unified under one principle.

Let VK denote the class of functions

that are analytic in the unit disc U, satisfy f′(z) ≠ 0 in U, and map U onto a domain with boundary rotation at most Kπ (for a definition of this concept, see [9]). V. Paatero [9] showed that f(z) ∊ VK if and only if

1.1