Search

1. Denote by G the group of the equation upon which depends the determi. nation of the curves of order n 3 having simple contact at 1 n ( n -3 ) points with a given curve C of order n having no double points. The case in which n is odd was discussed in the former paper (Transactions, January, 1902) and G was shown to be a subgroup of the group defined by the invariants 43, 04, , , * *, the latter group being holoedrically isomorphic with the first hypoabelian grouip on 2p indices with coefficients taken modulo 2. For n even, G is contained in the group H defined by the invariants 04' 069 * with even subscripts. JORDAN has shown (Traite, pp. 229-242) that H is holoedrically isomorphic with the abelian linear group A on 2p indices with coefficients taken modulo 2. The object of the present paper is to establish the latter theorem by a short, elementary proof, which makes no use of the abstract substitutions [al, 1 ., p, p1] of JORDAN, and which exhibits explicitly the correspondence t between the substitutions of the isomorphic groups.

In the paper with the above title, which appeared in these T r a n s a c t i on s, vol. 2 (1901), p. 428, I obtained a number of general theorems concerning the zeros of functions of the form 02 y' ol y, where y is a solution of a hornogeneous linear differelntial equation of the second order. There s a further general theorem of the same character which I should have included in that paper if I had discovered it at the time, and which I now give as VII' below, it being a counterpart to VII in the earlier paper. I also give three special applications of this general result, numbered VIII', VIII", VIII"', since they are counterparts of VIII. A part of VIII' was obtained by Fite in the Annals of M athematics for June, 1917, (see the closing lines of his article) by another method; and it was this special result of Fite's that suggested to me the much more general Theorem VII'. What follows should be regarded as an addendum to the former paper, to be inserted on page 434 at the end of ? 2. All references are to that paper, to which the reader should turn for an explanation of the notation and a statement of the restrictions placed on the functions. VII'. If none of the six functions

This paper deals with the construction of somep-groups of maximal class. Introduction. Let G be a group, G2 = [G, G] be the commutator subgroup of G and G,,I = [Gi, G] for i > 2. A p-group G of order pT is said to be of maximal class if IG: G21 = p2 and IGi: Gi+II = p for i = 1, . . ., n 1. In addition if G is of maximal class then GI is defined to be the largest subgroup of G such that [GI, G2] 2p, w 5 for the groups under consideration are known when p = 2 or p = 3. When p = 2, GI is cyclic, so G is metacyclic; when p = 3, G2 is abelian, that is G is metabelian, so these groups are also easy to classify [1, p. 82]. The structure of the groups is quite different when p > 5. If G is of order pf and G. is the first abelian member of the lower central series for G then, as we shall see, w may be as large as n/4. The other assumptions, G, is of class 2 and n > 2p, are made to keep commutator calculations from getting out of hand. The following conventions, assumptions, and results will be used throughout this paper: G is a p-group of maximal class and order pf where n > p + 2; GI, G2, ... are defined as above; w is the smallest positive integer such that GW is abelian; G is generated by s and s, where s E G G, and si E G -G2; si = [s1_1, s] for i = 2, 3, .. ., n; G, = for i = 1, 2, . . ., n -1; S = 1 for m > n. The symbol a(i, v) will always denote an integer such that 0 3. Those groups where w 3 then [G,-,, GJ] < G,-p+3. This was proved by Blackburn [1, p. 77]; it will also be proved here for it follows quite easily from results we have and need for other purposes. The relation [G,-,, GJ] < Gn_p+3 is equivalent to p-4 [Swl, W J~-II 5a(wqlNq q=O Received by the editors July 22, 1971. AMS (MOS) subject classifications (1970). Primary 20D15; Secondary 20F05.

The " universe of discourse " of the present paper is the class of all welldefined systems (K, R) where K is any class of elements A, B, C, ..., and R is any triadic relation. The notation R [ABC], or simply ABC, indicates that three given elements A, B, C, in the order stated, satisfy the relation R. Examples of such systems (K, R) are the following, of which example (a) is the most important: (a) K is the class of points on a line; AXB means that the point X lies between the points A and B. (b) K is the class of natural numbers; AXB means that the number X is the product of the numbers A and B. (c) K is the class of human beings; AXB means that X is a descendant of A and an ancestor of B. (d) K is the class of points on the circumference of a circle; AXB means that the arc A-X-B is less than 180O. (e) K is a class comprising four elements, namely, the numbers 2, 6, 6, an 6 A X Ce A 1V1) 74= A .T) anu O?o; LLa n mieans~ A =1 At D. It is obvious that these systems, and others like them, will possess a great variety of properties expressible in terms of the fundamental variables K and R. The object of this paper is to state clearly the characteristic properties of the type of system represented by example (a) above, by which this type of svstem is distinguished from all other possible systems (K, RI). In Section 1, we give a basic list of twelve postulates, due essentially to Pasch,t from which various sets of independent postulates will later be selected.

In recent years several papers [1; 3; 4; 5; 6; 7; 9; 10; 11] have appeared dealing with the problem of "Factorisatio numerorum," the number f(n) of representations of an integer n as an ordered product of factors greater than 1. As a result, the basic combinatorial properties of f(n) and the asymptotic behavior of its summatory function are well known. In this paper, I determine the asymptotic behavior of f(n) itself for squarefree n and use the result to determine a "normal" order of f(n) for all n. If n is written as a product of powers of distinct primes pi, then f(n) is evidently a symmetric function of the exponents of the pi's. If n is squarefree, all the exponents are 1 and f(n) can be considered a function of the single variable r, the number of distinct prime factors of n. It is therefore convenient to define, for positive integral r

In a paper with the above title published in these Transactions* there appeared a statement that "B(x) becomes infinite as x approaches x". In a letter, dated October 18, 1921, Professor Hans Hahn called attention to the fact that the proof of this statement, as implied by the context, although not explicitly mentioned, was insufficient, because it depended upon the nonvanishing of the quadratic form ,5 jij Cj cj, which had been proved to be definite by the aid of the fact that Z(x) + 0, and this condition does not hold at x1. It is the purpose of this note to supply a proof of the statement quoted above, upon which the validity of the concluding theorem of the earlier paper depends. I acknowledge my indebtedness to Professor Hahn for having pointed out the need for this supplemenitary proof. Starting with the definition of B(x) as given on page 435 and putting rij (x) = ikP (xI) Z(kj) (x), we have B (x) = j rij cj cj/Z. We shall show first that not every rij can vanish at x1 of the same order as Z(x). For, suppose that Z(x) = (x-xl)s Z1 (x), Z, (x1) + 0. If now every ij (x) had the factor (xx)s, then, since IRk((Xi)l Ot it would follow that every Z(Jk) (x) had it also. From the definition of the functions Z(Jk) (x), we derive the following system of equations

The purpose of this paper is to give a necessary and sufficient condition in order that the S limit set of a point consists of exactly one periodic orbit which is both S-Lyapunov stable and S-asymptotically stable. The general reference for most terms used in this paper is [2]. In this paper we assume (X, T, r) is a transformation group. X will always be assumed to be a uniform space where the Hausdorff topology in X is induced by the uniformity. T will be assumed to be generative, that is, T is isomorphic to Rm X I X C where R is the additive group of real numbers, I is the additive group of integers, m and n are nonnegative integers, and C is a compact abelian group [4]. S will denote a closed replete semigroup in T. The S-limit set of yEX, Sy, is defined bySv = n te,s Cl (ytS) [1 ]. DEFINITION 1. The orbit of a point yEX is said to be uniformly S-Lyapunov stable with respect to BCX provided that for each index a of X there is an index ,3 of X such that if weEBnyt'13 for some t' E T then wt Eyt'ta for all t E S.

1. Introduction. Finite non-Desarguesian projective planes have been known for all orders pf, where p is prime, n _ 2, and pn_ 9, except when p = 2 and n is a prime ? 5. In this paper a new class of projective planes is defined, having the orders 2n where n > 5 is not a power of two, thus establishing, in particular, the existence of non-Desarguesian planes of the missing orders. The new planes are coordinatized by semifields (sometines called division algebras, non-associative division rings, or distributive quasifields), which are algebraic systems satisfying the axioms for a field except with a loop replacing the multiplicative group. It is easy to show that finite proper semifields, i.e., semifields which are not fields, must have the orders pn where p is prime, n _ 3, and pn > 16. Proper semifields of these orders have been known to exist except as above, when p = 2 and n is prime; therefore the new systems show that proper semifields do exist for any order not excluded by simple arguments. A detailed treatment of the general theory of semifields and their relation to projective planes may be found in [3]. The manner in which these planes where discovered is perhaps as interesting as the planes themselves, since computers played a key role in the discovery. The author had received copies of two tables prepared by R. J. Walker (see [4]), which listed all commutative semifields of order 32 for which, if x is a generating element, x(x(x(x2))) = x + 1 or x2 + 1, respectively. There were 24 solutions in each table, and so it seemed plausible that a rule could be found yielding a correspondence between one set of solutions -and the other. A few hours of "cryptanalysis" did, in fact, result in the discovery of such a rule; and, since one of the solutions for the table x(x(x(x2))) = x2 + 1 was the field GF(32), the corresponding system for the other table could be written in a simple algebraic form [2]. After this, there was little difficulty showing that the same construction could be generalized to the construction of proper semifields of all orders 22k+1 with k > 1, and subsequent generalizations yielded the systems described in this paper. Therefore, we have an example in which

A stronger form of semi-aposyndesis is defined and used to characterize those compact, metric continua which are hereditarily locally connected. In [1], Bennett raised the question as to whether strongly aposyndetic continua are necessarily hereditarily locally connected. The purpose of this paper is to give an affirmative answer to the question and to prove the somewhat stronger theorem of the title. A continuum M is strongly aposyndetic if M is decomposable and whenever H and K are proper subcontinua such that M=HUK, then H and K are aposyndetic. The continuum M is strongly semi-aposyndetic if M is aposyndetic and whenever H and K are proper subcontinua such that M= HUK, then H and K are semi-aposyndetic. Since a strongly aposyndetic continuum M is easily seen to be aposyndetic it is not necessary to insist that M be aposyndetic in the definition. However, in the semi-aposyndetic case, requiring M to be decomposable does not insure that M is aposyndetic which it must be in order for the theorems of this paper to be valid. The method of proof will be to first show that an aposyndetic continuum in which every subcontinuum with nonvoid interior is semi-aposyndetic is necessarily hereditarily locally connected. Secondly, it will be proved that a strongly semi-aposyndetic continuum has the property that every subcontinuum with nonvoid interior is semi-aposyndetic. These two results together will complete the proof. Throughout, a continuum is to be a compact, metric, connected topological space. A continuum M is aposyndetic at p where p is a point of M if for every point q in M-p there exists an open set U in M and a subcontinuum H in M such that p E Uc Hc: M-q. If M is aposyndetic at p for each p in M then we say that M is aposyndetic. A continuum M is semi-aposyndetic at {p, q} where p and q are distinct points of M if there is Received by the editors July 15, 1971. AMS 1970 subject classifications. Primary 54F20, 54F50.

For some time now, very little has been known about abelian subgroups of pgroups.We shall try and remedy this situation in a series of papers, of which this is the first. One impetus for doing this was created by several natural conjectures which arose in problems in other areas of group theory. In this paper we shall study p-groups with respect to the existence and nonexistence of abelian subgroups which in one sense or another can be considered as large subgroups. Later work will deal with several other topics. Our first result is in a negative direction; we shall demonstrate the falsity of the conjecture that every group of order pn has an abelian subgroup of order pn/2. Previously, we announced [1] that the best one could hope for was abelian subgroups of order at least p*48n, but for odd primes we can greatly improve this result by an entirely different method.

Swingle [7]1 has given the following definitions. (1) A continuum M is said to be the finished sum of the continua of a collection G if G = M and no continuum of G is a subset of the sum of the others.2 (2) If n is a positive integer, the continuum M is said to be indecomposable under index n if M is the finished sum of n continua and is not the finished sum of n +1 continua. Swingle has shown [7, Theorem 2] that if n is a positive integer and the continuum M is indecomposable under index n, then M is the finished sum of n indecomposable continua. The author has shown [2, Theorem 1] that if n = 2 and the continuum M is indecomposable under index n, and G is a collection of n indecomposable continua whose finished sum is M, then G is the only such collection. In the present paper, it is shown that for a compact continuum, this theorem holds for any positive integer n. Also, there is given a necessary and sufficient condition that a compact continuum be indecomposable under 'index n. An indecomposable continuum can be described as a nondegenerate continuum which is indecomposable under index 1. If n =1, then in order that a continuum M be indecomposable under index n, it is necessary and sufficient that M contain n+2 points such that M is irreducible about any n+1 of them.3 Swingle [7] has shown that it is impossible, in a certain manner, to generalize this theorem. Theorem 3 of the present paper might be considered a generalization of the necessary condition of the above theorem. However, it is easily seen that the converse of Theorem 3 is not true. Theorems 1-5 are proved on the basis of R. L. Moore's Axioms 0 and 18. Hence these theorems hold in any metric space.4

Subsequently Leighton [3] proved that conditions (2) were sufficient without the restriction q be positive for sufficiently large x. In this paper we prove analogous theorems for the linear equation of order 2n (ry(n)) (n) + ( -1)n-lqy = 0. There will be no sign restrictions on the function q. For n= 1, the earlier results of Wintner and Leighton will be contained in our theorems. The results of this paper parallel earlier results of the author for a fourth-order equation with middle term [1]. Throughout r and q denote continuous, real-valued functions on a ray [a, oo) and r is assumed to be positive-valued. If the real-valued function y has n continuous derivatives and ry(n) has n continuous derivatives, then we define L(y) by

1. In 1878 JORDAN proved a theorem concerning linear homogeneous groups which may be enunciated as follows: Every such group G in n variables has an abelian self-conjugate subgroup F of orderf, and the order of G is )Xf, where X is inferior to a fixed number which depends only upon n.t The proof of this theorem is quite remarkable, the more so since the limit of X is not determined. The writer of the present article is not aware of any attempts that have been made since 1878 to find a limit to Xaside from the cases n 2, 3 4 -and he presents herewith some theorems which, in connection with some given by him in these Transactions, vol. 4 (1903), pp. 387-397, can be utilized to determine a number that X must divide, at least in the case of "primitive" groups. However, the chief object of the present paper is not this, but rather the presentation of some methods and theorems that are useful in the construction of the groups considered. As an illustration, the primitive groups in three variables are enumnerated at the end of the paper. The technical terms and phrases defined in the paper On the order of linear homogeneous groups, these T r a n s a c t i o n s, vol. 4, already referred to, will be retained here. As the present article is considered a continuation of this earlier paper -to which we shall hereafter refer by Linear groups I we shall begin with Theorem 8, meaning by the Theorems 1-7 those of Linear groups I. Unless otherwise stated, the substitutions used are linear and homogeneous in the variables concerned, and of determinant 1.

It is the object of this paper to determine the number of oscillations of a linear combination of the form (2) for the systems (3) and (4). From these results, an oscillation theorem for the solution up (x), corresponding to the pth characteristic number of (4), is obtained. Given the second order self-adjoint linear differential equation d du1 (1) dxL~~~K x,Xdx G (x,X) u 0 (1) ~~~dx [I z ) dZz,W and two linear combinations of a solution which does not vanish identically and its first derivative, (2) Li [ u (x,X) ] = ai (x,X) u (x,X) (i (x,X) K (x, ) ux (x ,X) (i = 1, 2), we shall impose the following conditions and shall assume that they -are satisfied throughout this paper: I. K(x,X), G(x,X), ai(x,X), i(x,X), aix(x,X), fix(x,X)t are continuous, real functions of x in the interval (X) (a _ x b) and for all real values of X in the interval (A) (-i 0 in (X, A) . III. For each value of x in (X), K and G decrease (or do not increase) * Presented to. the Society, Sept. 6, 1917. tfix (x x) a fi (x, x) 136

INTRODUCTION A prinicipal minor, or coaxial minor, of a determinant A is a minor obtained by striking out from A the same rows as columns. There are 2n1 principal minors of a determi-nant of the nth order, the determinant itself being included, but only n2 n +1 of them are independent.t For n = 1, 2, 3 the principal minors are all independent and for n = 4 the relations between them have been quite extensively studied.t For n>4 little has been published either as to which minors constitute an independent set or as to the relations between the minors. It is one of the purposes of this paper to determine several different types of complete sets of independenit principal minors of the general determinant of the nth order and to show how the elements of the determinant may be expressed in terms of the minors of an independent set. If we have a second determinant of the n th order with elements independent of the elements of the first determinant, there is a definite set of determinants obtained by replacing one or more columns of the first determinant by the corresponding column or columns of the second determinant. The set of principal minors of all such determinants is greatly enlarged over the set from the single determinant. It is the second purpose of this paper to determine several types of complete sets of independent principal minors of this enlarged set. There is also determined in this paper a complete set of independent principal minors of the determinants obtained when the above process is extended by adjoining to the original determinant more than one determinant of the nth order with independent elements.

In the second paper in this series [2; 3], it was demonstrated that under suitable topological conditions on a space S (which are in particular always satisfied for a manifold), the problem of reconstructing S from its group G of homeomorphism onto itself may be reduced to the problem of reconstructing the points of S from G. If S can be reconstructed then every automorphism of G is inner, and it was shown in this way that every automorphism of G is in fact inner when S is the unit interval; this is the theorem of Fine and Schweigert [I]. The present paper is devoted principally to proving the FineSchweigert theorem for the disc. Given a space S topologically equivalent to the interior of the unit circle in the plane, the point set of S will be canonically reconstructed from the group G of homeomorphisms of S onto itself. A principal tool is the classical theorem that if g is an orientation-preserving homeomorphism of finite order of the disc S onto itself, then S may be mapped homeomorphically on the interior of the unit circle in such a way that g becomes a rotation, [4]. Since a homeomorphism of odd order is necessarily orientation-preserving, it follows that an element of odd order (different from one) of the group G of homeomorphisms of S onto itself has precisely one fixed point, which will be called its center. If x is any point of S, let x' denote the set of all elements of G of finite odd order (different from one) having x for center. The collection S' of all sets x' is in canonical one-one correspondence with the point set of S. Our problem will be at an end if S' can be exhibited as a structure derived from the purely algebraic structure G. To this end it is sufficient to be able to decide within the structure of G when two elements of odd order (different from one) have distinct centers. This occupies the remainder of the paper. Henceforth, when an element of G is said to be of odd order, it will be understood that the order is different from one.

Introduction. Information about the integral representations of finite groups has been obtained to varying extents. For Z the ring of rational integers and G the cyclic group of prime order, the ZG-modules were studied by Diederichsen [3] and Reiner [11], who showed that there were finitely many indecomposable ZG-modules and determined them completely. The finiteness of the number of indecomposables in the case where G is cyclic of order p2 was shown, for p = 2, by Troy [16] and for any p by Heller and Reiner [5] and by Knee [8], while Oppenheim [10] and Knee [8] established the finiteness of the number of indecomposables for G cyclic, of square free order. Heller and Reiner [5; 6] and Jones [7] established that the number of indecomposable ZG-modules is finite if and only if all p-Sylow subgroups of G are cyclic of order at most p2. Here, as well as throughout this paper, we shall mean by a ZG-module one which is finitely generated and Z-free. In this paper we shall classify all finitely-generated S-free SG-modules where G is the dihedral group of order 2p, p an odd prime, and S is Z or Z2p the semilocal ring formed by the intersection of Zp and Z2, respectively the rings of p-integral and 2-integral elements in s the rational field. Z2p = {r/seQ: (s,2p) = 1}. Taking 0 to be a primitive pth root of unity, we shall denote by K = Q(9) the cyclotomic field of degree p — 1 over s and by K0 = Q(9 + 9~l) the real subfield of K. R0 and R shall be the integral closures of S in K0 and K, respectively. Letting I) denote the group of automorphisms of K with fixed field K0, we may form A the twisted group ring of h with coefficients in R. §1 of this paper is devoted to a characterization of R-projective A-modules of finite R-rank. The results of this section are then applied in the second section to show that there are precisely 7/t + 3 nonisomorphic, indecomposable SGmodules where h is the ideal class number of R0. In §3 it is shown that although a Krull-Schmidt theorem is not obtainable for SG-modules, invariants may be obtained which determine an SG-module up to Z2pG-isomorphism. The final section deals with projective SG-modules. Here an isomorphism is established

1. Jordan published his final contribution to the problem of the limit of transitivity of a substitution group which does not contain the alternating group in Liouville's Journal of 1895.4 Here he gave the interesting inequalities between n, the degree, and t, the muttiplicity of transitivity, of a t-ply transitive group, stated in the following theorem: Let n be the degree of a t-ply transitive group G of class >3. Then if t _ 8, nt _2a, a an integer> k-3 -log k/log 2, and k an integer such that 5?k 8.? In recent years Miller has given still another relation between n and t, namely, n > (4/25) (t + 2)2.11 In this paper inequalities similar to those of Jordan will be established. To obtain these it was necessary to study separately the t-ply transitive group whose subgroup that fixes t letters is of order a power of two and the t-ply transitive group whose subgroup that fixes t letters has its order divisible by an odd prime. The methods used in the study of the former group, though suggested by Jordan's 1895 paper, are new, while those used in the investigation of the latter group follow to a certain extent those of Jordan. The following theorems give the chief results of this study: Let the subgroup that fixes t letters of a t-ply transitive group of degree n and class>3 be of order 2m. Then if t_8

then Ea, is said to be absolutely summable by Rieszian means of order k and type X, or summable R, X, k| . The case X), n is of particular interest in this paper. Summability R, n, k| has been shown by J. M. Hyslop [3] to be equivalent to absolute Cesaro summability of order k, or summability I C, k| . One of the principal results shown by Obreschkoff was the consistency of the j R, n, k means; that is, he showed that summability |R n, kI implies summability R, n, k' , where k'> k. In this paper we introduce a method of absolute summability based upon the (a, ,B) method of summability defined by Bosanquet and Linfoot [1]. Just as the Bosanquet-Linfoot method generalized Riesz's arithmetic mean (R, n, a), the method given here will generalize absolute Rieszian summability R, n, a I. DEFINITION 2. A series Ea, is said to be absolutely summable (a, ,B), or summable a, f31, where a >0 or ao=0, f3>0, if for each sufficiently large C

where p is a prime =1 (mod e) and g is a primitive root of p, can be expressed in terms of the Jacobi sums of order e [12, (2.7)]. In 1935, L. E. Dickson published a series of three papers which reviewed and extended the theory of cyclotomy. In the first he gave relationships between the Jacobi sums of orders e, e < 6, e= 8 10, and 12 [3]. He discussed the sums of prime order and of order twice a prime, then treated explicitly e = 14 and 22, in the second paper [4]. In his final contribution he studied e = 9 and 18. (Sign omissions in formulas (37) were noted in [1]; an ambiguous sign in (44) was also resolved.) The last part of this paper is entitled

Let (7r) be the maximal ideal of the ring R of P-integral elements of an algebrai'c number field K, where P is a prime of K dividing the rational prime p. The natural homomorphism from R to K=R/(7r) induces a map S->S from the set of representations by matrices with coefficients in R of a finite group G into the set of representations of G in K. The lifting problem in modular representation theory is to determine whether for a given representation T of G in K there exists a representation S of G by R-matrices such that S= T. In this paper we introduce a notion of lifting projective modular representations from characteristic p to characteristic zero and show how this concept may be applied to the lifting problem. NOTATION. Throughout this paper G denotes a finite group of order I GI and K denotes an algebraic number field which is a splitting field for G. Let p be a rational prime and let R be the ring of P-integral elements of K, where P is a prime of K dividing p. Let (r) be the maximal ideal of R and set K=R/(r). K is a finite field of characteristic p which is a splitting field for G. For aCR, set d=a+(ir)CK. If A = (aij) is a matrix with entries in R (R-matrix) we denote by A the matrix (aij). By a linear representation of G in a field L we shall understand a homomorphism from G into GL(m, L) for some m. By a projective integral representation (resp. projective modular representation) of G in R (resp. K) we mean a map T of G into GL(m, K) (resp. GL(m, K)) satisfying T(1)1=lm, T(g)T(h)=a(g, h) T(gh) where a(g, h) CR (resp. K) and T(g) has entries in R (resp. K) for all g, hCG. a is called the factor set associated with T. If a(g, h) =f(g, h) = 1 for all g, hC G, T is a linear integral representation (resp. linear modular representation). We identify linear representations with projective representations having trivial factor sets. We refer the reader to [3] and [7] for the relevant theory. DEFINITION. Let T be a projective modular representation of G in K and let a be the associated factor set. T is projectively liftable if there exists a projective integral representation S of G in R with factor set A such that S(g) = T(g) for all gCG. If a(g, h) =,3(g, h) = 1 for all g, hCtG (i.e. S and T are linear representations), we say that T is liftable.

In this paper we define and discuss some of the properties of certain subsets of a continuous curve,t which we call the arc-curves. Probably the most interesting property of these subsets is that whenever the arc-curve is closed, it is itself a continuous curve. The arc-curves are intimately connected with the theory of separation of points in continuous curves, since if a closed set separates a pair of points in the arc-curve of the pair of points with respect to a given continuous curve, the closed set separates the points in the given continuous curve. We define a basic set of a continuous curve as any set whose arc-curve is the continuous curve itself. Necessary and sufficient conditions are given in order that a subset of a continuous curve be a basic set and in order that a basic set be irreducible. It is assumed that all point sets of the paper lie in a two-dimensional euclidean space.t Notation. Wherever a symbol H is used to denote a point set, the symbol H7 denotes the point set consisting of the points of H plus all limit points of H. If u and v are points of a simple continuous arc ae, the symbol uv denotes the subarc of ae with u and v as end points if ui v, and if u = v, the symbol uv denotes the single point u(=v). Hereafter we shall use the term "arc" to mean "simple continuous arc." If uv is an arc with end points u and v, the symbols , denote uv-u, uv-v, uv-u-v respectively. If A and B are sets of points, the symbol A c B means A is a subset of B, but it is not implied that A is a proper subset of B.

In a recent paper,' T. Szele proved that a division ring D is orderable if and only if the additive and multiplicative semigroup S generated by the nonzero squares of elements of D does not contain the zero element of D. The present paper extends this result to a domain of integrity K. Let us denote by K* the set of nonzero elements of K. The domain of integrity K is said to be orderable if and only if there exists an additive and multiplicative semigroup P (the positive elements) contained in K* such that K*=PUJ(-P). If K does not have a unit element, then there exists a unique minimal domain of integrity K having a unit element and containing K.2 It is not too difficult to show that K is orderable if and only if K is orderable. For this reason, we assume henceforth that K has a unit element. An element a of K* is called even if there exist elements a,, ,an in K* such that a is a product of the 2n elements a,, an, a,, * *, an in some order. We denote by S the additive semigroup generated by the even elements of K*. The theorem we wish to prove is as follows.

If a compact Lie group acts effectively on a manifold, then local cross-sections to the orbit structure can be constructed. They are called slices. We prove that if a compact group acts effectively on a manifold, but it is not a Lie group, then a slice cannot exist at every point. Introduction. Recent papers of R. F. Williams have focused new attention on the Hilbert-Smith conjecture: If a compact group G acts effectively on a manifold, then G is a Lie group. Much interest has centered around properties which arise if one assumes that the conjecture is false, and that the p-adic group A, acts effectively on a manifold M. The purpose of this paper is to show that if such an action occurs, then for any reasonable definition of a slice, there is at least one point on the manifold at which a slice fails to exist. DEFINITION. Let (X, G) be a compact G-space, and let xeX. A slice at x is a subset Y of X containing x and satisfying: (i) Whenever g E G and y E Y, g(y) E Y?g E Gx. (ii) If Q is an open neighborhood of eCG/G. which serves as a local cross-section to the cosets of G. at the identity of G, then (g, y)-?g(y) defines a homeomorphism of QX Yonto Q(Y). Moreover, Q(Y) is open in X. REMARK. Equivalent definitions of a slice for the case where G is a compact Lie group, may be found in Montgomery and Yang [4], and in Chu [1], and Palais [6], for noncompact Lie groups. DEFINITION. Let Gk denote a cyclic group of order pk, and let gkA; GA be a fixed generator. For each pair of integers i, j>: 0 with i?j we let cij::Gi -Gj be the homomorphism which arises by sending gi-?gj. Note that cii is the identity map on Gi. With the positive integers as directed indexing set, it is easily verified that the collection of groups and homomorphisms { Gi, cij } forms an inverse system. The group A, = lim, { Gi } is called the p-adic group. Its topology and group operation are those which are induced from the product X5Gi, and from the standard theorems on inverse limits we find that Ap,is abelian, Hausdorff, and compact. Received by the editors March 5, 1970. AMS 1969 subject dassifications. Primary 2240, 5480.

The first part of this paper deals with established results concerning complete balanced Howell rotations for an even number of teams. The point of view used in this paper gives rise to independent proofs of known properties of these rotations. Further, the terminology of Room squares readily lends itself to this study and the existing theory of Room squares enables us to give a short proof of the important Berlekamp and Hwang [1] result. Two new constructions are given in section 5. A Room square (RS) or a Room design of order 2n, where n is a positive integer, is an arrangement of 2n elements (symbols) in a square array of side 2 n 1 such that (a) each of the ( 2 n 1) 2 cells of the array is either empty or contains an unordered pair of distinct elements, (b) each of the 2n objects appears precisely once in each row and in each column, and (c) each unordered pair of distinct objects occurs in exactly one cell of the array. For a description of results about RS's and an extensive bibliography, the reader is referred to the paper by Mullin and Stanton [9]. References for some of the more recent results are available in Horton's paper [5]. Now let us consider duplicate bridge tournaments and Howell rotations (progressions, movements). The ideal situation in a duplicate bridge tournament for 2n teams (partnerships) is to arrange that

Affine or projective Hjelmslev planes are called 1-uniform (also strongly 1-uniform) if they are finite customary affine or projective planes. If n > 1, an n-uniform affine or projective Hjelmslev plane is a (finite) Hjelmslev plane U with the following property: for each point P of ?, the substructure n1 p of all neighbor points of P is an (n 1)-uniform affine Hjelmslev plane. Associated with each point P is a sequence of neighborhoods lp C 2p C ... C np = 2X. For i k, ('i)j and (i k -).k are isomorphic. Then if 2I(i) denotes (i?)i_ 1(1), q... , I(n) is a periodic sequence of ordinary translation planes (all of order r) whose period is divisible by k. It is proved that if T 1 . 9 Tk is an arbitrary sequence of translation planes with common order and if n > k, then there exists a strongly n-uniform translation Hjelmslev plane U of width k such that I(i) T. for i < k. The proof of this result depends heavily upon a characterization of the class of strongly n-uniform translation Hjelmslev planes which is given in this paper. This characterization is given in terms of the constructibility of the n-uniform planes from the (n 1)-uniform planes by means of group congruences. Introduction. Hjelmslev planes are generalizations of customary affine and projective planes in which distinct lines may intersect in more than a single point. Throughout this paper, we will refer to a Hjelmslev plane as an H-plane. One calls a finite H-plane 1-uniform if it is a customary affine or projective H-plane: Received by the editors May 14, 1971. AMS (MOS) subject classifications (1970). Primary 05B25, 50D35; Secondary 05B30.

It is shown that if x, y are generators of the finite group G such that xv = yg = (xy)" = 1, where p, q, n are integers > 1, (p, q) = 1, and xy is of true order n, then the order It = nt of G satisfies n < pqt2. This result is used to show that if Fis a Fuchsian group of genus 0 generated by 2 elliptic elements of coprime order and with 1 parabolic class, then F possesses only finitely many normal subgroups having a given number of parabolic classes. Introduction. Let r = LF(2, Z) be the classical modular group, so that r is the free product of the cyclic group {T} of order 2 and the cyclic group {U} of order 3, where T = (_? 1), U = (o -1), and S= TU= (1 1) is parabolic. Let G be a normal subgroup of r of index ,u and level n (the least positive integer such that Sn E G). Then if t is the number of parabolic classes of G, t = ,u/n. In [3] the conjecture was made that there are only finitely many normal subgroups of r having a given number t of parabolic classes. This conjecture was proved by L. Greenberg in his paper [2], who showed that n < 6t3, so that It ? 6t4. In connection with his work on automorphisms of closed Riemann surfaces, R. Accola [1] proved a number of group-theoretic lemmas about finite groups generated by 2 elements one of which is of period 2. A consequence of his work is the improvement of the inequality above to n < 6t2 always, and n ? t2 if rIG is not abelian. In this paper we apply Accola's method with some modifications to the more general situation of any finite group G generated by 2 elements, whose orders are assumed relatively prime for our purpose. We obtain a number of inequalities connecting the order of G with the orders of the elements and their products. The results are used to show that Fuchsian groups of genus 0 with two elliptic generators of coprime order and with one parabolic class have the same sort of property as r with respect to the parabolic class number. The author is indebted to the referee for correcting an error in the original version of the paper. Received by the editors November 2, 1970. AMS 1970 subject classifications. Primary 20H10, 20D99; Secondary IOD05.

where a is an arbitrary constant. This result is less satisfactory than that concerning the number p(n) of partitions (unrestricted) of n, since in the latter case the error term approaches zero with increasing n. Recently Rademacher(4) obtained an equality for p(n). The object of the present paper is to find an equality for q(n). The work of this paper is a straightforward application of Hardy-Ramanujan's method with two modifications. These modifications are Kloosterman's sum and Rademacher's "Farey dissection of infinite order." The present method may also be applied to find the explicit formula for

In [4] Y. Tao proved that if h h is a fixed point free involution of S 1 × S 2 {S^1} \times {S^2} , then ( S 1 × S 2 ) / h ({S^1} \times {S^2})/h must be homeomorphic to either M 1 = S 1 × S 2 {M_1} = {S^1} \times {S^2} , or M 2 = K 3 {M_2} = {{\mathbf {K}}^3} , or M 3 = S 1 × P 2 {M_3} = {S^1} \times {{\mathbf {P}}^2} or M 4 = P 3 # P 3 {M_4} = {{\mathbf {P}}^3}\# {{\mathbf {P}}^3} . In this paper we extend this result to free actions of Z 2 n {Z_{{2^n}}} on S 1 × S 2 {S^1} \times {S^2} , showing that, for n > 1 , ( S 1 × S 2 ) / Z 2 n n > 1,({S^1} \times {S^2})/{Z_{{2^n}}} must be homeomorphic to either M 1 {M_1} or M 2 {M_2} .

Although neither the result nor the e-transformation is new, a new order for the successive transformations is prescribed. From this follow some interesting properties which in turn imply the result. Let A, B and C be subsets of a finite abelian group G, with a, b and c the respective elements. Suppose further that A +B=C. DEFINITION. Let J ={x:x+Bc C}. We say A is full whenever A=A. Notice that A c'A and A +B=C. DEFINITION. Let AC={x:x+Cc C}. It is obvious that 0 AC, C+AC' C, AA c AC (in fact AA =AC if A is full). Also C+Sc C implies that SGAC. Hence AC+AC=AC, so that AC is a subgroup of G. We now use the usual Dyson e-transform. H. B. Mann's e-transform would serve equally well with only the obvious modifications in the proof, and indeed the proof was originally done using that transform. For e in A, a full set, and 0 E B another full set, A A' =A u (B + e), B B'= B (A -e), C C' = A' + B'. Also let A* = {a:a E A', a 0 A}, B*= {b:b eB, b r B'}. We have the following usual properties: PO. C, c C, P1. B*+e=A*, P2. IAI+IBI=IA'I+IB'I, P3. OeB'. Received by the editors January 13, 1972 and, in revised form, February 15, 1973. * This article has not been proofread by the author because the Amer. Math. Soc. was unable to locate him. The address given at the end of the paper is the last address given by the author. ? American Mathematical Society 1974

In this paper we establish the existence of a best Lp approximation, 1 ⩽ p ⩽ ∞, to a given function f∈Lp( D , where D ⊂ Rm is a bounded domain, from the family Vn(S) of all nth order exponential sums in m independent variables for which the corresponding exponential parameters lie in the closed set S ⊆ C. In so doing we extend the previously known existence theorem which corresponds to the special case where m = 1 and D is a finite interval.

In the present paper the waiting system G/G/1 is investigated. We form the functionals (3.1) with the service-time distribution B or the inter arrival time distribution A or K (see (1.1)).By means of them it is possible to define a haif order relation of random variables or distributions functions, respectively. We consider rhree systems G/G/1 and estimate - with reference to this relations - the distributions A, B, K above and below by the corre- sponding distribiitions of the two other systems considered as known. Then upper and lower bounds of the semiinvariants of the stationary waiting time (i e. the mean waiting time, variance etc.) of the given system are obtained.

If S,, is a Sylow p-subgroup of the symmetric group of degree pn, then any group of order pn may be imbedded in Sn. We may express Sn as the complete product' C o C o ... o C of n cyclic groups of order p and the purpose of this paper is to show that any Sylow psubgroup of a classical group (see ?1) over the finite field GF(q) with q elements, where (q, p) = 1, is expressible as a direct product of basic subgroups En-C O C O ... o C (n factors), where Z is cyclic of order pr. (We assume always that p ;2.) Since C may be imbedded in S., we see that n is imbedded in Sn+r-l in a particularly simple way. The above r is defined by the equation q -1 =pt *where qI is the first power of q which is congruent to 1 mod p and * denotes some unspecified number prime to p. The case r = 1 is therefore of frequent occurrence, and then clearly SnSn Professor Philip Hall was my research supervisor in Cambridge (England) during the years 1949-1952 and it is a pleasure to acknowledge here my indebtedness to him for his generous encouragement.

ship of these secondary obstruction classes when m > 2n. In this paper, we shall study the higher obstructions to sectioning a special type of fibre bundles such that 72(F) is the first nontrivial homotopy group of the fibre and the dimension of the next homotopy group of F is much higher than 4. Let F be a homogeneous space of a compact connected semisimple Lie group G. Suppose that the first nontrivial homotopy group of F is m2(F) which is free. Let Q' be a bundle over X with fibre F and group G and whose primary obstruction vanishes. We will construct a principal torus bundle F, -- F, a central extension G1 of G by the torus, and a set of weakly associated bundles {4 over X with fibre F1 and group G1. Under the projection p: 4 -, secondary and tertiarv obstructions for 4 corresponding to primary and secondary obstructions for ('s (cf. Theorem (4.3)). We shall call this method enlarging the fibre and the group. Thus, an enlargement may be used to reduce the order of an obstruction. By this method, we rederive Kundert's formula for the secondary obstructions when the fibre is a complex projective space CP'-1 and PU(n) is the structural group(2). We will also derive new formulas for the third obstructions when F = CPI'- 1 and for the secondary obstructions mod 2 to finding an orientable 2plane sub-bundle of an orientable vector space bundle (cf. Theorems (6.1) and (7.1)). The author extends his cordial thanks to Professor N. E. Steenrod for his constant encouragement, suggestions and criticisms. He equally thanks Professors J. C. Moore, J. W. Milnor and Dr. P. F. Baum for many helpful discussions.

This paper investigates the relationship between the p-length, l p ( G ) {l_p}(G) , of the finite p-solvable group G and the number, a p ( G ) {a_p}(G) , of orbits in which the subgroups of order p are permuted by the automorphism group of G. If p > 2 p > 2 and a p ( G ) ≦ 2 {a_p}(G) \leqq 2 , it is shown that l p ( G ) ≦ a p ( G ) {l_p}(G) \leqq {a_p}(G) . If p = 2 p = 2 and a 2 ( G ) = 1 {a_2}(G) = 1 , it is proved that either l p ( G ) ≦ a p ( G ) {l_p}(G) \leqq {a_p}(G) or G / O 2 ′ ( G ) G/{O_{2’}}(G) is a specific group of order 48.

1. In his first memoir on analysis situst Poincaredefined a number N(Fk, Jin_k) which had previously been considered, at least in special cases, by Kronecker. With certain conventions as to sign this number represents the excess of the number of positive over the number of negative intersections of a k-dimensional circuit Fk with an (n k) -dimensional circuit F,i_ when both are immersed in an n-dimensional oriented manifold. The purpose of the present paper is to show how to calculate this number when the manifold is defined combinatorially as a collection of cells and the circuits are composed of sets of these cells; and to show how the matrices which represent the inter. sectional relations between the k-circuits and the (n k) circuits depend on the matrices of orientation of the manifold. We also define certain modulo 2 intersection numbers and discuss the matrices coinnected with them. The terminology and notations of the Cambridge Colloquittm Lectures on Analysis Situs (New York, 1922) will be used without further explanation, and the references not otherwise indicated will be to that book. 2. Let a manifold Mn be given as the set of all points of a complex C(. Let C' be a complex dual to Cn constructed as explained on page 88 by means of a complex Cn which is a regular subdivision both of Cn aind of C'. Every k-cell 0 of Cn has a single point PI (cf. p. 85) in common with a single (n -k)-cell of C' whiclh is called bjy-k. Our first problem will be to assign a positive or negative sign to the intersection of 0 with b.k. In order to do this, we suppose M, to be orienited as explained in Chapter IV and that all cells, circuits,. etc., are oriented. Moreover, in the regular complex Cn, in which each i-cell is uniquely determined by its i + 1 vertices, the orientation of the i-cell will be denoted by the order in which its vertices are written, and the following two conventions will be followed: (1) if AO A1 .. Ak denotes a given oriented k-cell (k 1, 2, ..., n) any even permutation of Ao A1 .. Ak denotes the same oriented k-cell and any odd permutation denotes its negative; (2) the oriented (k 1 )-cell A1 A2 ... Ak iS positively related to the oriented k-cell Ao A, * * Ak.

for t in [a, c], p, q, g are real continuous functions on [a, c] and A, B, C are real 2X2 matrices; in the determinate case, has a nonsymmetric Green's function if BF-zO. The purpose of this paper is to generalize his result to higher order problems. Consider problems: (*) Y' = F Y with A 1 Y(a) +Bi Y(b) + Ci Y(c) = 0 where A 1, B1, C1 and k X k matrices, k > 1, Y is a k-dimensional vector and F = (fij) satisfies

* Presented to the Society at the San Francisco meeting, April 30, 1904. Received for publication July 12, 1904. t BURNSIDE, On the Representation of a Group of Finite Order as an Irreducible Group of Linear Substitutions, etc., Proceedings of the London Mathematical Society, November, 1903, pp. 117-123. The quantities Xi have been called weights by the author in two papers published in these Transactions, vol. 4 (1903), p. 387, and vol. 5 (1904), p. 310. t Cf. equation (true for transitive groups) given by BURNSIDE, P r o c e e d i n g s o f t h e L o ndon Mathematical Society, NMarch, 1903, p. 122, bottom. 461

1. The theorem which I now propose to prove is THEOREM XIII. Let q be any positive integer greater than 5, and let p be any prime greater than 2q 2; then the degree of any primitive group G that contains a substitution of order p and degree qp but none of order p and of degree less than qp does not exceed qp + 4q 4; if G is not triply (doubly) transitive its degree does not exceed qp + 4q 6 (qp + 4q 7); the order of G is not divisible by p2. 2. In the two former papers in these Tr a n s a c t i o n s t under this same title, and to which references are indicated by the Roman numerals I and II in parenthesis, a proof and a slight extension were given (for q greater than unity) of the theorem which Jordan stated in the Memoir on Primitive Groupsinthefirstvolumeof the Bulletin of the Mathematical Society of France, page 175: Let q be a positive integer less than 6; p any prime greater than q; the degree of a primitive group G that contains a substitution of order p on q cycles (without including the alternating group) can not exceed qp + q + 1. To this one may add that when p is greater than q + 1, the order of G is not divisible by p2. 3. The memoir at the end of which this theorem is given is devoted to the larger question of the corresponding limit for the degree of G when q is not confined to numbers less than 6. Jordan's general result may be stated thus: Let q be any positive integer, p any prime greater than 2q log2 q + q + 1; the degree of a primitive group G that contains a substitution of order p on q cycles (without including the alternating group) can not exceed qp + 2q log2 2q. 4. This is supplemented in certain directions by the theorem:T If a primitive group of class greater than 3 contains a substitution of prime order p and of degree qp (q less than 2p + 3), it includes a transitive subgroup of degree not greater than the larger of the two numbers qp + q2 q, 2q2 p2. In particular, if q is less than p + 2, the degree of G, when it is simply transitive, is not greater than qp + q2 q.

1. The paper to which this is a sequel had to do only with those multiply transitive groups of class ,(>3) in which at least one substitution of degree , is of even order. t Among other results, it was there proved that the degree n of triply transitive groups of class ,(>3), which contain a substitution of even order on , letters, does not exceed 2,. Triply transitive groups of class A and degree 2, exist. This measure of success was due to the simplicity of structure of diedral rotation groups of class , generated by two similar substitutions of degree , and order 2. Up to the present time no better limit than that of Bochert, n_3,u(,u>6), has been found to apply to these triply transitive groups of class , >3) in which all substitutions of degree , are of odd order. t But it will here be proved that if such a group contains a substitution of order pc (p an odd prime) on , letters

One of the most important systems of groups is that which is composed of all the groups which may be defined by the orders of two generators (s1, s2) and the order of their product. It is known that these three orders determine a finite group only when one of them is unity, when two of them are equal to 2, or when one is 2 while the other two are one of the following three pairs of numbers: 3, 3; 3, 4; 3, 5.t The first of these sets of orders define a cyclic group and may be regarded as trivial. When the order of two among the three operators si, 82, 81 92 is equal to 2, the group { s1, 82 }, generated by s, , 82, is the dihedral rotation group whose order is twice that of the third operator. This useful category has recently been generalized under the headling, The groups generated by two operators which have a common square4t This generalized category could also be defined as the set of groups generated by two operators such that the square of one of them is equal to the square of their product. The object of the present paper is to complete this kind of generalization for the dihedral rotation group and to extend it to the other groups of genus zero. The resulting groups have a two-fold interest in view of their close contact with the important system of groups of genus zero and their elementary structures. It is believed that a complete list of these groups will prove very useful in many investigations. Our object is to study all the groups which result when one of the three conditions which may be used to define a group of genus zero is preserved while the other two are replaced by a single one of an elementary type. For instance, the dihedral rotation groups are defined by the conditions ?

To the theory of multiply transitive groups Bochert has contributed formulae for the upper limit of the degree of a group when the number of letters displaced by a substitution in it is known, f But »for simply transitive primitive groups a corresponding theorem of such elegance does not exist. J However, in the present paper is offered a theorem with some claims to simplicity. It may be stated thus : If a simply transitive primitive group contains a substitution of prime order p and of degree pq ( q less than 2p 43 ), its degree is not greater than the larger of the two numbers qp 4q2 — ?, %q2 —p2Also when the given substitution of prime order has more than 2p 42 and less than p2 cycles or when p is 2 or 3, our results are capable of fairly concise expression ; but when q exceeds p2—1 and p is greater than 3 the corresponding upper limit is unfortunately a more complicated function of p and q. The subject matter of this study is not restricted to the simply transitive primitive groups, but it is proposed to find an upper limit for the degree of some transitive subgroup of a primitive group known to contain a substitution of order p on qp letters. It seems unnecessary to state that a simply transitive primitive group has no transitive subgroup of a lower degree.

It has recelntly been proved that in every non-cyclic group of order pt", p being an odd prime, the number of cyclic subgroups of order pg, /3> 1, is always a multiple of p, and that this number is of the form 1 + p + kp2 when /3 = 1.t From this it follows alrnost directly that the number of cyclic subgroups of order pg in any group (G) is always of the form kp whenever the Sylow subgroups of order ptm in G are non-cyclic, and that the number of subgroups of order p in such a G is always of the form 1 + p + kp2. j ' hen p = 2, both of these theorems have exceptions. The present paper is devoted to an exhaustive study of the exceptions of the former theorem. Since the cyclic groups are so elementary we shall confine our attelntion to the noln-cyclic groups of order 2m. Moreover, every group of even order contains an odd

where D* is the unit disc u2 +?v2 < 1 and T* is continuous on D*. If the Lebesgue area of this second surface is denoted by A (T*), then the question arises: How should the two representations T and T* be related in order for A (T) to equal A (T*) ? 0.2. It is well known that A (T) =A (T*) if T can be obtained from T* by a change of parameters, in precise terms, if there exists a homeomorphism H(D) =D* such that T(p) = T*H(p) for every point p in D. More generally, A (T) =A (T*) if T is F-equivalent (equivalent in the Frechet sense) to T*, that is, if for every positive number e there exists a homeomorphism H1(D) =D* such that the distance in Euclidean 3-space between the points T(p) and T*He(p) is less than e for every point p in D. Since two F-equivalent representations yield the same Frechet surface, this is merely a restatement of the fact that the Lebesgue area is independent of the choice of representation for the surface (see 0.1). In the sequel we shall write T-T*(F) to indicate that T and T* are F-equivalent. We shall also speak of the F-invariance of A (T), meaning that T-T*(F) implies A (T) = A (T*). 0.3. The principal result of this paper is to establish the relation A(T) =A (T*) in a still more general situation; namely, when T is K-equivalent to T* (in symbols, T-T*(K)), that is, if T and T* possess simultaneous monotone-light factorizations T=LM and T* =LM*, where Mand M* are continu

In this paper propositions will be established concerning the collection of all the cut points of a given planet continuum. It will be shown that there does not exist an uncountable collection of mutually exclusive subcontinua of a given continuum M each of which contains at least one cut point of M. With the aid of this fundamental theorem it is shown, among other things, that the set of all the cut points of a continuum M which are irregular points of M is necessarily countable and, indeed, that all save a countable number of the cut points of any continuum M are points of Menger order two of M.

This paper completes the analysis of the Jacobi sums of order 22 outlined by L. E. Dickson. Furthermore, it is shown that a prime p p of the form 22 f + 1 22f + 1 has the binary quadratic decomposition 4 p = u 2 + 11 v 2 4p = {u^2} + 11{v^2} and that certain Jacobi sums can be evaluated in terms of u u and v v , where u u satisfies u ≡ 9 ( mod 1 ) 1 u \equiv 9\pmod 11 .

The question arises as to the convergence of the development (I) to f(x) or-what is the same-the behavior of rnf (x) in (II) for n -? o. The case (a, b) (-1 11), p (x) 1 leads to Legendre's polynomials; it has been treated by Professor D. Jackson.t In this paper we follow the method given by Professor Jackson in order to investigate the convergence of the development (I) involving Tchebycheff's polynomials in general. Hereafter-, the interval (a, b) is supposed to be finite, and f(x) to be continuous on (a, b). 1. rnjf(x) expressed as a definite integral. We obtain easily, using the formulas for Ai, fb fn rn,f (x) == f (x) -Jp (y)f (y) 2; So (x) soj (y) dy, 0

The totality of 4X4 matrices with coefficients in GF [22], determinant unity, and which permute among themselves the vectors of dimension 4 having 4 co-ordinates nonzero marks of GF [22], or 2 zeros and 2 nonzero marks of the field, forms a simple [1, p. 285, problem 9, part (4)]' (multiplicative) group, G, of order 26 .34 5. It will be demonstrated that G is isomorphic with no multiply transitive permutation group. After submitting the manuscript, the author received a communication, quoted in part, "The referee has observed that if one is willing to use Frame's table (Duke Math. J. vol. 2) for the characters of this group, then the result can be derived in a few lines as follows. The degrees of the irreducible representations are 1, 5, 6, 10, 15, 20, 24, 30, 40, 45, 60, 64, 81. If the group had a doubly transitive permutation representation of degree n, then n would divide the order of the group and n -1 would be a degree of an irreducible representation. Moreover, the character of this irreducible representation has to be rational and its values must be > -1. This eliminates all degrees except n -1 = 15. However, the character in question has the value 3 for an element of order 4 and values -1 for all elements of order 2. This would mean that the element of order 4 leaves 4 letters fixed while its square changes all letters." In the remainder of the paper the counter-example will be established independently of characters: In order that G be representable as at least doubly transitive on n symbols, it is necessary that n!>26 34 5, and n(n-1) be a divisor [1, p. 141, Theorem II] of 26 34 .5. Thus n must be 9, 10, 16, or 81. The cases 9 and 10 are easily disposed of, for 10! is divisible by only the fourth power of 3. Hence, if G were represented on 9 or 10 symbols, then each of its Sylow subgroups belonging to 3 must be a Sylow subgroup of the symmetric group; such a subgroup contains a cycle of 3 symbols, which generates a primitive group of degree 3. But a doubly transitive [1, p. 160] group is primitive; and a primitive [3, p. 92, Theorem II ] group of degree n containing a primitive subgroup of degree m is at least (nn-m+1)-fold transitive. This is the desired contradiction.

where J and N are positive integers, and the X's are real or complex constants. The present paper gives the chief results of a study of this problem. In order to add precision to the problem, and in order to exclude certain extreme cases which are of minor interest, it is assumed (1) that we do not have XoN=lX= * =XJN; (2) that we do not have XOn =X,= * *n =Xin:i$O for any n

In a group G, we let S(x) denote the normal subsemigroup generated by x and S'(x) = S(x)YJe. Following Fuchs [2], an element x of G is said to be generalized periodic (GP) iff eES(x) and nongeneralized periodic (NGP) otherwise. A group is said to be an 0* group iff every partial order on G can be extended to a full order. Ohnishi [9] considered the following group-theoretic condition: (i) For all x, y, t in G, if xES(t) and yES(t), then S(x)f\S(y) $0. He proved that a group G is an O* group iff every nontrivial element of G is NGP and G satisfies condition (i). In this paper we will study groups which satisfy condition (i) and will also consider two other conditions, one of which is equivalent to (i) and another which is implied by (i). We remark first that if G satisfies (i), it is known and easily verified that any homomorphic image of G also satisfies (i). Thus a homomorphic image of an 0* group is an 0* group iff every nontrivial element is NGP.