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

Tahara [1] has concretely determined the conjugate classes of finite subgroups of GL(3, Z). The group W5 in his list of groups of order 24 is in fact of order 12 and consists of the same matrices as the group W6 in his list of groups order 12. Hence, there are only 10 conjugate classes of subgroups of order 24 in GL(3, Z) and the total number of conjugate classes of finite subgroups is reduced to 73.

Let G toe a 2-transitive permutation group of a set !! of npoints and let P be a Sylow p-subgroup of G where p is aprime dividing \G\ . If we restrict the lengths of the orbitsof P , can we correspondingly restrict the order of P ? In theprevious two papers of this series we were concerned with thecase in which all P-orbits have length at most p ; in thesecond paper we looked at Sylow p-subgroups of a two pointstabiliser. We showed that either P had order p , orG > A , G = PSL(2, 5) with p = 2 , or G = M of degree 12with p = 3 . In this paper we assume that P has a subgroup Qof index p and all orbits of Q have length at most p . We2conclude that either P has order at most p , or the groupsare known; namely PSL(3, p) S G5 PGL(3, p) ,ASL(2, p) £ G 5 AGL(2, p) , G = PrL(2, 8) with p = 3 ,G = M with p = 3 , G = PGL(2, 5) with p = 2 , or G > A2with 3p - n < 2p ; all in their natural representations.Let G be a doubly transitive permutation group on a set ft of npoints and let P be a Sylow p-subgroup of G where p is a primedividing \G\ . The previous two papers [9, JO] were concerned with thesituation in which P has no orbit of length greater than p . We showedessentially that either G contains the alternating group or P has orderp . The general problem is the following:Received 21 May 1975.

Erdös [1] has shown that, if the square-free numbers in ascending order be denoted by s1, s2, … , sn, … , then for 0 ≦ γ ≦ 2as x → ∞. In this paper we shall extend this result by proving that the asymptotic formula in fact holds for the wider range 0 ≦ γ ≦ 3.Similar results have been obtained previously by the author in respect of both the sequence of numbers expressible as the sum of two squares and also sequences of numbers relatively prime to given large integers, although the method used here differs from that of the earlier papers [2; 3].

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.

A detailed probabilistic treatment is given of a birth-and-death process proposed by Williams (1969) in which the elements of the process bear up to s inanimate marks. Equations for the second-order moments of the process, and approximate marginal univariate solutions, are derived. The exact bivariate solution is given for the case s = 1. For general s the variance of the mark population is also derived. MARKED BIRTH-AND-DEATH PROCESS; MULTIVARIATE LAGRANGE DIFFERENTIAL EQUATIONS; MULTITYPE BRANCHING PROCESS Introduction In a recent paper Williams (1969) has considered the following process. We have a population of elements developing as a linear birth-and-death process and each element bears a number of inanimate marks. When a marked element dies then it and its marks are removed from the population; when a marked element divides, its marks segregate independently and without loss, each with probability I into one of the two daughter elements. It is assumed that marking an element does not affect its birth and death rates (cf. Clifford and Sudbury (1972)). Williams' process was proposed as a model of an experiment conducted on bacteria (the elements) infected by phage (the inanimate marks), by Meynell (1959). (The bacteria are lysogenic with respect to the phage, which means that the phage do not interfere with the behaviour of the bacteria.) His object was to use the marking process in order to separate the birth and death rates of the bacteria; if we denote birth and death rates by A and p respectively, then the mean-value measures on the bacterial population (assumed to develop as a linear birth-and-death process) will estimate the difference (A p). As the marks are inanimate, however, they will only be affected by p and so mean-value measures on the mark population enable us to estimate p (see Williams (1969)). The papers of Meynell and Williams taken together provide a most interesting example of a virological experiment and the probability model of that experiment. Received in revised form 10 August 1973 Research sponsored by a Science Research Council grant and a World Health Organisation research traineeship. 423 This content downloaded from 207.46.13.180 on Thu, 08 Sep 2016 05:01:40 UTC All use subject to http://about.jstor.org/terms 424 BYRON J. T. MORGAN Williams considered the mean-value solution of his model; this was adequate for Meynell's experiment in which population sizes were large. Here we shall give a fuller probabilistic statement of the model and make some contributions to its solution. Two differences from Williams' work lie in the adoption of an upper bound, s, to the number of marks any element can have, and the assumption (except where stated) of birth and death rates independent of time. We 'shall assume general initial conditions which, in Meynell's experiment, correspond to the distribution of phage over bacteria once these have been initially exposed to a phage population. Williams took this distribution to be Poisson; the distribution of phage over bacteria has, however, been the subject of much probabilistic study (e.g., see Gani (1965) and Morgan (1971)). Finally, let us note that the value s = I, on which we shall particularly concentrate, probably gives the most realistic description of Meynell's experiment. 2. The probability model The model is a continuous-time Markov process with discrete states. Let N,(t) be a random variable denoting the number of elements with i marks at time t, i = 0, 1,2, ..., s. Let n = (n0,n1, l , n.., ), Iu = (Uo, u1, *,us), N(t) = (No(t), N(t), --,N,(t)), and suppose N,(O) = ai, i = 0, 1,2, --, s. Let P(n; t) = P(N(t) = n), then corresponding to death we have the transitions, (No, ..., Ni, ---, N,) (No, --, Ni 1, , N,) at probability rate pNi, i = 0, 1,2, ,s, while corresponding to birth we have the following transitions (No, N,) ,(No + 1, .., N,) at probability rate, 2 =1 N 2-' +,ANo, (No, , Ni ,N2i,..., X (No, Ni + 2, , N2iI, at probability rate, This content downloaded from 207.46.13.180 on Thu, 08 Sep 2016 05:01:40 UTC All use subject to http://about.jstor.org/terms On the distribution of inanimate marks over a linear birth-and-death process 425 where we use [is] to denote the integral part of is, and (No, " , Ni, * * *, Nk i Nk, "', Ns) -*(No, ..., Nj + 1, ..., Nk-i + 1, ..., Nj 1, ..., Nl) at probability rate 2N 2-k' i = 1,2,...,C[(k 1)], k = 3,4,...,s (smaller k give transitions already considered above). It is readily verified that the sum of the transition rates is (A + yp) X = oN,, as required in the overall birth-and-death process. Introducing the operator Di, which increases the argument ni of a function of n by 1, we then deduce that the Kolmogorov forward equations (for all possible n) are, d S S (1) +(n; t) P(n; t)(A+ t)1 (n2 + D,P(n;t)(n +1) dt ==0 1=0

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.

In the preceding paper Bose, Shrikhande, and Parker give their important discovery of the disproof of Euler's conjecture on Latin squares. In this paper we show that their results can be strengthened to imply that N(n), the maximal number of pairwise orthogonal Latin squares of order n, tends to infinity with n. In fact there exists a positive constant c, such that N(n) > nc for all sufficiently large n.Our proof involves no new combinatorial insights, but is based entirely on a number-theoretical investigation of the following inequality due to Bose and Shrikhande.

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

The problem has been solved by Stoll for the case p ( n 1 ) [6] and for the case where n(r, A) is bounded [9], [10] and [11]. In the first part of this paper the case p = 0 is treated. The second part of the paper is devoted to the case 0 < p < (n 1) with the additional condition that the given analytic set A in Vis regular on an ( n 2 -p) -d imens iona l projective subplane of the infinite plane of the projective closure of V. The author wishes to thank Professor Wilhelm Stoll for his advice and encouragement.

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.

This paper applies the constant population approximation to the study of epidemics which involve more than a single type of infective. An example of this would be a situation in which both clinically infected individuals and subclinically infected individuals or carriers are present. We derive equations for the expected numbers of clinically infected individuals and carriers at any time t for the model with zero latent period and infectious periods having negative exponential distributions. From these equations we derive conditions under which a unimodal incidence curve can result, and expressions for the expected total epidemic size. The equations describing the course of an epidemic are analytically intractable except in certain simple cases, [1]. When one examines the reason for the mathematical difficulties that arise, one finds that the principal stumbling block is the assumption of a finite population of susceptibles. It can be argued, however, that in modern societies epidemics very rarely menace an entire population, and that observed epidemic sizes are usually much smaller than the total susceptible population. This is due to a variety of reasons, among which are public health measures and good communication facilities available to a modern society. In consequence it is reasonable to try to bypass some of the mathematical difficulties inherent in the theory of epidemics by assuming a susceptible population whose size does not change through the course of the epidemic. Bailey [2], Morgan [3], and Williams [4] have discussed in some detail the theory of epidemics using the constant population approximation, although such an approximation was used by several authors earlier, [5], [6], [7]. The theory developed so far has dealt mainly with epidemic processes in which there is only a single type of infective and a single type of susceptible. Recently Gart [8] has considered a model for epidemics involving more than one type of susceptible. It is the purpose of this paper to analyze the development in time of epidemics which involve more than a single type of infective individual. This Received in revised form 3 January 1967. 257 This content downloaded from 157.55.39.78 on Mon, 20 Jun 2016 07:20:57 UTC All use subject to http://about.jstor.org/terms 258 HUGH M. PETTIGREW AND GEORGE H. WEISS problem, in which there may be several types of infectives, is suggested by diseases in which carriers are important, [9]. In the present paper we derive equations for the expected numbers of carriers and clinically infected individuals at any time t, in the case of a zero latent period and negative exponential distributions of infectious periods. From these equations we will give the conditions under which an epidemic arises from the introduction of a bearer of the disease, i.e., the conditions under which the reporting curve increases initially. These conditions are of some theoretical interest since Bailey has shown that in the infinite population approximation to an epidemic with only one class of susceptibles and one class of infected individuals, no initial increase in the reporting curve can occur. Finally, we derive expressions; for the expected total epidemic size. Let us consider a homogeneously mixing population which, in the present approximation, can be characterized by four parameters, yi(t), y,(t), zi(t), zc(t). These are, respectively, the number of clinically infected individuals, the number of carriers, the cumulative number of removals of infectives, and the cumulative number of removals of carriers, all evaluated at time t. The probability that a single susceptible individual will become clinically infected in (t, t + dt) is assumed to be fli(y, + yc)dt, and the probability that a susceptible will become a carrier is assumed to be Ic(yi + ye)dt, where both l's are assumed to be constant and it is assumed for simplicity that carriers and infectives are equally infectious. The constancy of these p's is the principal assumption of the present theory; in the more detailed stochastic theory discussed by Bailey [1] the f's are proportional to the number of susceptibles as well. In the simplest model, we assume that the latent period is zero, i.e., that newly infected individuals are immediately capable of infecting others, and that the infectious period is a random variable with a negative exponential distribution. The rate parameter appearing in the distribution for clinically infected will be denoted by vi and that appearing in the distribution for carriers will be denoted by v,. In order to describe the stochastic process we introduce a set of probabilities p(r,,r2, r3, r4, t) defined by p(r, t) = Pr{y,(t) = r, yc(t) = r2, z(t) = r3, zc(t) = r4}. By our assumptions the p(r, t) satisfy ap = fl (r1 + r2 1)p(r, 1,r2,r3, r4, t) + fl(r, + r2 -1)p(r1,r2 1,r3, r4, t) (1) + vA(r1 + 1)p(r + 1,r2, r 1,r4)+ vc(r2 + 1)p(r, r2 + 1, r3,r4 1) [(fl + ic + vi)rl + (fl + flPc + ve)r2] p(r, t). This content downloaded from 157.55.39.78 on Mon, 20 Jun 2016 07:20:57 UTC All use subject to http://about.jstor.org/terms Epidemics with carriers: the large population approximation 259 The moment generating function (2) M(01, 02, 03 , 0,t) = M(O, t) = E{eo'Y'+2yYc+03zi+042C} therefore satisfies aM M = [f#,(ee I1) + flc(e02 1) + vi(ee, +03 1)] (3) aM + [fl(ee' 1) + flc(e02 1) + vc(e-02+4 1)] The interesting features of the development of the epidemic can be determined by analyzing the properties of the mean values Lpi(t) = E{yi(t)}, pc(t) = E{yc(t)} (4) ow(t) = E{z,(t)}, wc(t) = E{zc(t)}. The equations for these mean values are easily determined by equating coefficients of the O's on both sides of Equation (3). They are Pi = (/01v-i)it?+fliltc (5) Pc = flcpi+ (fc v)Pc

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

It is shown in this paper that total biorthogonal systems have the same cone if and only if they differ at most by rearrangement and by positive scalar multiplication. A connection is demonstrated between this result and work done by R. E. Fullerton in which he characterized the existence of an unconditional basis in terms of the existence of certain type cones. The paper is concluded by generalizing the first result to the situation in which two biorthogonal systems have cones which induce order isomorphic orderings. l Definitions and notations* In this paper we will assume that

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.

Our aim is to present some global results about the set of maximal abelian subgroups of the symmetric group Sn. We shall show that certain properties are true for “almost all” subgroups of this set in the sense that the proportion of subgroups which have these properties tends to 1 as n → ∞. In this context we consider the order and the number of orbits of a maximal abelian subgroup and the number of generators which the group requires.Earlier results of this kind may be found in the papers [1; 2; 3; 4; 5]; the papers of Erdös and Turán deal with properties of the set of elements of Sn. The present work arose out of a conversation with Professor Turán who posed the general problem: given a specific class of subgroups (e.g., the abelian subgroups or the solvable subgroups) of Sn, what kind of properties hold for almost all subgroups of the class?

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.

A subgroup V of G is called an j in jector of G if Vc~ N is F-maximal in N (that is, Vn N is a maximal ~-subgroup of N) for each subnormal subgroup N of G. It is known that the ~-injectors of a group G form a conjugacy class of subgroups 1,2]. If ~-injectors are always normal then ~ is called a normal Fitting class. In I-1] and [-3] some work has been done to classify normal Fitting classes. It has been shown in these papers that to any normal Fitting class there corresponds an object called a Fitting pair (f, A) defined as follows: A is an Abelian group and to each finite soluble group G there exists a homomorphism fG: G~A such that for all N ~ Gf~ = f~lN, where f~lN is the restriction of fG to N. Each such pair defines a Fitting class ~ given by G ~ if and only if fG(G)= 1, 1,Satz 3.1, 1.]. We use this to construct a Fitting class which contains $3, the symmetric group on 3 letters but does not contain the dihedral group of order 18. For some time it had been undecided whether the Fitting class generated by S 3 was the class consisting of all 3-groups extended by 2-groups. For a general survey see the paper of Cossey read at the conference on group theory held in Canberra, Australia 1973. The construction will actually provide uncountably many normal Fitting classes for all of which, the injectors have index 2. We will define and prove the existence of these classes by using Fitting pairs. Let f2 be the set of powers of odd primes and let 0 E f2. Let A be the cyclic group of order 2 and define, for any finite soluble group G

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.

Introduction. It is well known that the number p(n) of unrestricted partitions of a positive integer n can be expressed by a convergent series (see [9]). In this paper we shall be concerned with the number p (n; a, M) of partitions of n into positive summands congruent to ? a modulo Mi, where a, M are integers with M ? 2. This partition function p (n; a, M) has been treated for certain special values of M; namely, M 2 by Hua [2], M==5 by Lehner [5], M=6 by Niven [8], and M = p (p a prime > 3) by Livingood [6]; a convergent series representation of p (n; a, M) being obtained in each case. The main object of the present paper is to derive a convergent series for p (n; a, M) in which M assumes general values. We may suppose without loss of generality that 0 1 and d I n, it reduces to p (n/d; a/d, M/d). Further we remark that the case M =2 (partitions into odd summands) is equivalent to the case M = 4, i. e., p (n; 1, 2) -p (n; 1, 4). Consequently it suffices to consider the case M ? 3. We apply the Hardy-Ramanujan method with modifications due to Rademacher [10]. In order to utilize this method, however, it is necessary to solve the following two subsidiary problems: The first problem is to find a suitable transformation equation for the generating function of p (n; a, M). The usual method of contour integration seems to be rather complicated in our case. However, we can show, in a simple way, the existence of the transformation equation, the proof of which being based on a certain functional equation recently obtained by the author [3]. This will be done in Section 1 of this paper. The second is to get a non-trivial estimate for a certain exponential sum which is introduced as a consequence of the transformation equation. We shall

In this paper we apply methods of G. Higman, outlined in [2], to determine on which 2-groups Q the Suzuki groups Sz(q) can act as a group of automorphisms with an element of order 5 acting fixed point freely. We find that Q is elementary abelian and we determine precisely the action of Sz(q) on Q. To state what this action is, we need some definitions. In his paper [3], Suzuki defines the group G(q), here denoted by Sz(q), as generated as a subgroup of GL(4,q) by a certain collection of matrices. We let M be a 4-dimensional vector space over GF(q) affording this representation of Sz(q). If N is a module over the group-ring GF (q) Sz(q) (here GF(q) is the field of q elements), then we say that the action of Sz(q) on N is natural if N is isomorphic as GF (q) -Sz(q) module to a direct sum of copies of M1. If N is an elementary abelian 2-group admitting Sz(q) as a group of operators, then we say that the action of Sz(q) on N is natural if N is isomorphic, as GF (2) Sz (q) module to a GF (q) Sz (q) module on which Sz (q) acts naturally. With this notation, the following is a precise statement of the main result of this paper.

The following note may be considered as an addendum to the paper by me on pp. 42–47 of this volume of the Proceedings . In that paper it is shown how to inscribe in a triangle ABC, a triangle DEF, such that the perpendiculars to the sides of ABC, drawn through the points D, E, F, shall be concurrent in a point P. This is done by constructing on each of the sides of ABO a triangle similar to DEF; then O the point of concurrence of the three lines joining the vertices of ABC to the vertices of these triangles is the point “inverse” to P. The question, then, naturally arises, What must be the shape of the triangle DEF in order that the point P may be one of the Brocard points, and, as a consequence, O the other one? and the answer is easily seen to be that DEF must be similar to ABO. Hence the following construction:—

The list of the primitive groups of the first thirteen classes prepared by JORDAN has long been known to be defective in the case of class 12; in fact, JORDAN states explicitly that the calculations for that case, having been gone through but once, may very well contain errors.* In spite of the considerable advances in substitution theory in recent years, the complete a priori determination of all the groups of this class still involves a good deal of labor. In order therefore not to intrude too far upon the space of this journal, the writer avoids a redetermination of the primitive groups of class 12 on less than 21 letters.1 In a series of memoirs and in his lectures Professor MILLER has gone over this ground without use of the list of the groups of class 12. He has thus checked the results of MiSS MARTIN, of JORDAN, and of Mliss BENNETT on the degrees 18, 19, and 20, respectively, according to whiclh there is no primnitive gr'oup of class 12 on either of these three degrees. The method here followed will be much the same as that used by the author in treating the classes 6, 8, and 10, with the advantage of having at hand the processes employed in the paper entitled "On the Limit of the Degree of Primitive Groups." This last paper, as well as that "On Mlultiply Transitive Groups," and that "On the Order of Primitive Groups," will be used freely without specific reference.t In addition to the 25 primitive groups of class 12 of degree less than 18, there are four others, one of degree 27, two of degree 28, and one of degree 36.

In 1925 Menger f introduced the term "curve" for those compact connected spaces which have the property that for every point of the set and every e> 0 there exists a neighborhood of diameter less than e which contains the point and has a totally disconnected I boundary. A point of such a set is said to be regular if it is possible to choose these neighborhoods in such a manner that its boundary is finite, while it is of order n if for every e we can find the neighborhood of diameter less than e whose boundary consists of exactly n points. In a subsequent paper he shows that in a regular curve, i. C., a curse containing only regular points, there is corresponding to every point of order n a set of n arcs having the point in common and otherwise distinct.? It should be noted that every regular curve of Menger is also a continuous curve,? that is, it is conilected im kleinen at every poinlt. Of course there are continuous curves which are neither regular nor even curves in the sense defined above. In this paper is colisidered a plane continuous curve in which there are two points a and b which have the property that there are in the continuous curve exactly n arcs frorn, a to b which have no points other than a and b in comrnmon while it is impossible to find a similar set of n + 1 arcs from a to b. Under these hypotheses it is proved that there are exactly n points in the curve whose omission separates a from b in the curve. This theorem provides a sort of converse to the theorem of Menger mentioned. The precise converse of the theorem of this paper is an easy consequence of the theorem itself, and

In a previous paper 2 the writer defined the notion of a simply connected set and showed how it was useful in proving a number of separation theorems. Our purpose here is to extend this notion to multiple connectivity and derive some further results for the plane. The abbreviation S. C. will hereafter be used in referring to the paper mentioned above. Let n be a positive integer. A connected set M is said to be multiply connected of order n, or n-ply connected, if (1) for each pair of points A and B of M and any relatively closed subset L of M that separates A from B in M there exists a subset of L, consisting of n or less components, which separates A from B in M, and (2) there exists at least one pair of points A and B of M and a relatively closed subset L of M that separates A from B in M such that every subset of L which separates A from B in M has at least n components. This definition becomes the criterion for a connected set to be n-ply connected in the weak sense if " separates " is replaced throughout by " weakly disconnects." 3'

A point p of a compact metric continuum M is said to be a point of finite degree * of M provided that for each e > 0 there exists an uncountable family G of neighborhoods of p in Mll each of diameter < E anid each having a finite set for a boundary and such, that for any two neiborhoods U, V of G we have either U C V or V C U. Similarly, p is said to be of degree a provided ac is the least cardinal number such that for each e the neighborhoods of G can be chosen so that the boundary of each of them is of power < ac. (We shall be concerned principally with the case where a is an integer.) If X is a given subset of M, then p will be said to be,of finite degree or of degree ac in M relative to X provided that, in addition to satisfying the conditions previously stated, the neighborhoods of G can be so chosen that the boundary of each of them is a subset of X. These notions concerning the degree of a point should not be confused with the analogous but distinctly different notions concerning the order or index of a point as introduced by Menger and Urysohn. Both concepts will be used in this paper, but careful reading should enable one to avoid confusing them. A continuum, is said to be of finite degree provided each of its points is a point of finite degree. In this paper we shall show, among other results, that such continua may be identified with the continua previously studied by the author t which have the property that any subcontinuum contains uncountably many local separating points : of the given continuum.

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 1927, Redfield [13] wrote a remarkable paper on combinatorial theory in which he anticipated several enumeration results [4, 11, 12, 14]. We first became acquainted with this paper while studying the book on group characters by Littlewood [10], which references Redfield's paper. This is the only paper Redfield ever wrote. lRedfield's "decomposition theorem" was studied by Foulkes [3] using group characters to formulate the result. In a footnote, Foulkes mentioned some of Redfield's enumeration results. Our object in this expository paper is to give Redfield proper credit for his combinatorial discoveries, while making his methods more available by recasting them in contemporary notation and terminology [6, 7, 8, 9]. In particular, we display explicitly the basic group-theoretic result, which we call "Redfield's Lemma," a generalization of the well-known lemma of Burnside [1]. 1. Permutation groups. Let A be any permutation group of order in I AI acting on the set X of d objects. Let jk(a) be the number of eveles of length ic in the disjoint cycle decomposition of a permutation a. Let us denote d variables by a1,a2, , ad. Then the cycle indeX 2 Z (A) is the following formal sum, a polynomial in these variables: 1 d (I)~~~~~ Z(A ) =n -I a7, 1(a 1)m EA k=1

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

It is shown that every maximal partial order in a torsionfree abelian-by-nilpotent group is isolated. The same is true for an ordered polycyclic group. Examples are given to show that maximal partial orders need not be isolated in torsionfree polycyclic groups, nor in ordered centre-by-metabelian groups. l Introduction. In ([3], p. 209, Problem 3) Fuchs suggested the problem of investigating those groups in which every maximal partial order is isolated. In this paper we show the dichotomies produced by groups having the above property in the class of solvable groups. We call P a partial order on a group G if it is a normal subsemigroup of G such that Pf) P"1 = {e}. P is a maximal partial order if it is a maximal normal subsemigroup subject to P ft P~ι = {e}, and P is isolated if gneP implies g e P for all g e G and n > 0. Since every partial order on a group G can be extended to a maximal partial order using Zorn's lemma, the class of groups in which every maximal partial order is isolated is the same as that in which every partial order can be extended to an isolated partial order. We shall adopt the notation used by Hollister in [7] and denote this class of groups by /*. Clearly, every /* group is torsion-free since {e} itself is a partial order. Our main results are as follows. THEOREM 1. Every torsion-free abelian-by-nilpotent group is in Γ.

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.

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.