Our aim in this paper is to fill one gap left in (1) and to prove that if H is a finite collineation group of F5, the free plane generated by a finite open configuration of rank 9, then |H | ≦ 12. Alltop has shown that |H | ≦ 24 and that there exist finite collineation groups of F5 which have order 12, so that the argument in this paper shows that |H| ≦ 12 is the best estimate which can be given. In (1), Alltop has completely settled the question for Fn, n ≠ 5. The notation of this paper will generally be that of (1).

Most of the arguments used here will consist of case analyses of degenerate planes of ranks 7 and 8 and will be sketched rather than given in detail. The one theorem of some interest in this paper, other than the main result, is the following theorem which yields some information about the collineation group of F4 (π2 in the notation of (4)).

Our aim in this paper is to prove inequalities of the form

1

or

2

for all real values of the parameters α, β and all non-negative (in some cases all positive) xi. Obviously, an is finite in all cases, and we shall show that An is finite if α and α + β are both non-negative. In all cases, we obtain sharp values of the constants an, An (when finite), as well as bounds for these constants, and their behaviour as n → ∞. In case a < 0, we naturally consider only positive xi, otherwise the xi may be non-negative. Although we always write xi ≧ 0 in the following, this should be read as xi > 0 in case α < 0; similar remarks apply to the parameter t introduced below.

In this paper we deal with the number of zeros of a solution of the nth order linear differential equation

1.1

where the functions pj(z) (j = 0, 1, …, n – 2) are assumed to be regular in a given domain D of the complex plane. The differential equation (1.1) is called disconjugate in D, if no (non-trivial) solution of (1.1) has more than (n – 1) zeros in D. (The zeros are counted by their multiplicity.)

The ideas of this paper are related to those of Nehari (7; 9) on second order differential equations. In (7), he pointed out the following basic relationship. The function

1.2

where y1(z) and y2(z) are two linearly independent solutions of

1.3

is univalent in D, if and only if no solution of equation(1.3) has more than one zero in D, i.e., if and only if(1.3) is disconjugate in D.

In (9), Stasheff showed that a loop space ΩX is homotopy-commutative if and only if the map e∇ : ΣΩX ∨ ΣΩX → X may be extended to ΣΩX × ΣΩX, where ∇ is the folding map and e:ΣΩX → X is the map whose adjoint is the identity map of ΩX. In (5), Ganea, Hilton, and Peterson showed that this criterion does not dualize. Our aim in this paper is to give a reformulation of Stasheff's criterion, which is equivalent to it, but in a form which does dualize. In the course of the paper, we shall discuss why Stasheff's criterion, in its original form, does not dualize. In (5), the authors have, of course, given an explicit counterexample to the dual of StashefFs criterion in its original form.

1. For an entire function , let M(r, f), μ(r, f), and v(r, f) denote the maximum modulus, the maximum term, and the rank for |z\ = r, respectively. Also, let

and λ(r) the lower proximate order relative to log M(r, f). For the properties of the lower proximate order, we refer the reader to the paper by Shah (1).

2. We prove the following theorems.

THEOREM 1. For an entire function

where μ(r, f1) and M(r, f1) correspond to fl(z), the derivative of f(z), provided (n + l)Rn > nRn+1, when L(f) > 1.

In this paper, some contributions are made to the theory of algebraic equations over the rational field with conditions imposed on the Galois group. In § 1, for a given abstract group G all faithful permutation representations Ḡ are obtained, and it is shown that if one of them is the group of some equation with splitting field K, then any of them is the group of some equation, also with splitting field K. The method of proof enables us to construct an equation having as group a given faithful permutation representation Ḡ of a prescribed group G if we are given an equation having as group some faithful representation of G. In § 2, equations having nilpotent group are considered, non-normal extension fields are discussed, and a canonical form is obtained for the roots of non-normal irreducible equations; this form is used to characterize fields and equations with nilpotent groups.

f(n, k) sera l'entier maximum t tel qu'il existe un graphe G ayant les propriétés suivantes:

• (a) G possède n sommets;

• (b) le nombre chromatique de G est égal à k;

• (c) Gest minimal par rapport à la propriété (b); c'est-à-dire, la suppression ede n'importe quelle arête rend G(k – 1)-colorable;

• (d) il existe t sommets indépendants de G, c'est-à-dire dont nulle paire ne se joigne par une arête.

Un graphe sera k-minimal s'il possède les propriétés (b) et (c). Puisque les graphes 3-minimaux sont tous des polygones impairs, il s'ensuit que f(n, 3) = [n/2] (n = 3, 5, 7, …), Il y a quelque temps T. Gallai a posé la conjecture:

1.1

M. Simonovits a réfuté l'inégalité stricte dans (1.1) en prouvant que

1.2

A variety (primitive class) is a class of abstract algebras which is closed under the formation of subalgebras, homomorphic images, and products. For a given variety we shall call a function μ*, which assigns to each algebra a natural number or ∞, denoted by μ*(A), a variety invariant if for every natural number n the class of all with μ*(A) ≦ n is again a variety. In this paper, a general method of finding variety invariants for the variety of all modular lattices will be developed. This method will be based on the concept of a quotient tree of a modular lattice. As examples of variety invariants we shall define, using the general result, the primitive length and the primitive width of modular lattices.

In this paper we shall be concerned with the algebraic structure of certain rings of functions meromorphic on a non-compact (connected) Riemann surface Ω. In this setting, A = A(Ω) and K= K(Ω) denote (respectively) the ring of all complex-valued functions analytic on Ω and its field of quotients, the field of functions meromorphic on Ω. The rings considered here are those subrings of K containing A,which we term A-rings of K. Most of the results given here were previously announced without proof (15) and are contained in the author's doctoral dissertation (16), completed at the University of Illinois under the direction of Professor M. Heins, whose encouragement and advice are gratefully acknowledged.

All rings considered in this note have an identity element, and all R-modules are unitary.

Bass (2) defined a left perfect ring as a ring R satisfying the minimum condition on principal right ideals. A commutative ring R is perfect if and only if R is a direct sum of finitely many local rings whose radicals are T-nilpotent. Therefore, the commutative perfect rings with finite global projective dimension are just the direct sums of finitely many commutative fields, and hence they trivially satisfy the minimum condition for all ideals. However, in the non-commutative case, even hereditary perfect rings are not necessarily right or left artinian (cf. Example 3.4).

Each left perfect ring R has only finitely many idempotent (two-sided) ideals (Corollary 2.3), where the ideal X of R is called idempotent, if X = X2. Hence, it makes sense to consider minimal idempotent idealsof the left perfect ring R, i.e., ideals of R which are minimal in the set of all idemponent ideals of R.

H. Behr and J. Mennicke (1) have proven that the group PSL(2, p) can be presented by the following system of generators and relations:

1

From this presentation, it follows that the three relations

2

for the same generators S and T suffice if p > 3, p ≠ 17. If p = 3, it is well known that the relations S3 = 1, T2 = 1, and (ST)3 = 1 define PSL(2, 3). For p = 2, the relations S3 = 1, T2 = 1, and (ST)2 = 1 define PSL(2, 2). For p = 17, the three relations

3

will suffice.

Indeed, the group G, with generators S, T and defining relations (2), contains the subgroup 〈Sp〉 in its centre.

A function f, analytic in the unit disk, is said to have finite Dirichlet integral if

1

Geometrically, this is equivalent to f mapping the disk onto a Riemann surface of finite area. The class of Dirichlet integrable functions will be denoted by . The condition above can be restated in terms of Taylor coefficients; if f(z) = Σanzn, then if and only if Σn|an|2 < ∞. Thus, is contained in the Hardy class H2.

In particular, every such function has boundary values

almost everywhere and log |f(eiθ)| ∊ L1(dθ).

The zeros zn of a function must satisfy the Blaschke condition

and f(s) = B(z)F(z), where F(z) has no zeros and

is the Blaschke product with zeros zn; see (5).

Capital letters denote sets and the cardinal of A is |A|. Greek letters always denote ordinal numbers and, unless stated otherwise, small latin letters denote non-negative integers. The symbol [A]r is used to denote the set {X: X ⊂ A; |X| = r} of all subsets of A with relements. If A is a simply ordered set with the order relation <, then the order type of A with this ordering is written as tp <A or simply as tp A when there is no ambiguity about the intended order relation. A graph G = (A, E) is an ordered pair with A as the set of vertices and E ⊂ [A]2 as the set of edges. In particular, if A is simply ordered, we call G a graph of type tp A. A complete subgraph of G = (A, E)is a set B ⊂ A such that [B]2 ⊂ E; a set C⊂ A is independent if [C]2 ∩ E = ∅.

In this paper we shall classify all finite groups in which the centralizer of every involution has a normal 2-complement. For brevity, we call such a group an I-group. To state our classification theorem precisely, we need a preliminary definition.

As is well-known, the automorphism group G = PΓL(2, q) of H= PSL(2, q), q= pn, is of the form G = LF, where L = PGL(2, q), L ⨞ G, F is cyclic of order n, L ∩ F = 1, and the elements of F are induced from semilinear transformations of the natural vector space on which GL(2, q) acts; cf. (3, Lemma 2.1) or (7, Lemma 3.3). It follows at once (4, Lemma 2.1; 8, Lemma 3.1) that the groups H and L are each I-groups. Moreover, when q is an odd square, there is another subgroup of G in addition to L that contains H as a subgroup of index 2 and which is an I-group.

In a previous paper (5), I constructed a class of translation planes, called generalized André planes or λ-planes, and discussed the associated autotopism collineation groups. The main question unanswered in (5) is whether or not there exists a collineation η of a λ-plane Π which moves the two axes of Π but does not interchange them.

The answer to this question is “no”, except if Π is a Hall plane (or possibly if the order n of Π is 34) (Corollary 2.8). This result makes it possible to determine the isomorphisms between λ-planes. More specifically, let Π and Π′ be two λ-planes of order n coordinatized by λ-systems Qand Q′, respectively. Then, except possibly if n = 34, Π and Π′ are isomorphic if and only if Q and Q′ are isotopic or anti-isotopic (Corollary 2.13). In particular, Π is an André plane if and only if Q is an André system (Corollary 2.14).

Let p(z) be a complex polynomial of degree less than or equal to n. Generalizing the well-known Bernstein theorem, Szegö (3) has shown that

We shall give a partial generalization of this result.

THEOREM. Let p(z) be a polynomial of degree at most n. Let R be the radius of the largest disc contained in G = {p(z): |z| < 1}. Then

Since R ≦ max|2|=1 |Re p(z)|, we obtain Szegö's result, but with a worse constant. It would be interesting to see whether it is possible to replace the constant e by 1. If so, Rzn would be an extremal for all n, and another extremal for even n.

The proof is based on the following result of Ahlfors (1) (cf., e.g., 2, p. 321). This result corresponds to the estimate f or the Landau constant.

In the Amitsur-Kurosch theory of radicals in rings (2), an important problem is to determine the relationship between the radical of a ring and the radical of each of its ideals. The first result on this problem was by Amitsur who proved that if β is a hereditary radical in the sense that ideals of β-radical rings are β-radical, then for each associative ring R and ideal I of R, β(I) = I ∩ β(R), where β(R) denotes the β-radical of R; see (2).

Later, Suliński, Divinsky, and the author proved that if β is any radical and R is an associative or alternative ring, then β(I) ⊆ I∩ β(R) for each ideal I of R; see (3). Since every hereditary radical β has the property β(I) ⊇ I ∩ β(R), this result provided another proof of Amitsur's theorem and extended that theorem to alternative rings. Of course, this raises the question of whether Amitsur's theorem is true for Lie or Jordan rings.

Pólya (3) has shown that if bnis a sequence of algebraic integers and is a rational function, then so is . This result was generalized by Uchiyama (5) who showed that one may replace the assumption that the bn are algebraic integers by the assumption that the bn lie in a finitely generated submodule of the complex numbers, and by the author (1) who showed that if p is a non-zero polynomial with complex coefficients and if bn is a sequence of algebraic integers such that is a rational function, then so is . Our aim in this note is to give a common generalization of all of these theorems.

In this paper, certain results of Bing (1) and myself (2) are extended. It is well-known that a chainable compact metric continuum must be a-triodic (contain no triods), hereditarily unicoherent (the common part of each two subcontinua is connected), and each subcontinuum must be chainable. Our principal result states that a compact metric continuum M is chainable if and only if M is a-triodic, hereditarily unicoherent and each indecomposable subcontinuum of M is chainable. Some condition on the indecomposable subcontinua of M seems essential, if we consider the dyadic solenoid, 5, which is indecomposable, a-triodic and hereditarily unicoherent. Indeed, each proper subcontinuum of S is an arc. However, S is not chainable, since it cannot be embedded in the plane.