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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 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

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

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.

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.

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

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

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.

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.

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

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.

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.

* 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.

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

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

Sp(a) = {fp(z) = f(zP)l/p : f(z) e S(a)}. Extremal problems for the coefficients in the power series expansions of functions in Sp(O) and powers of such functions were recently investigated by J. T. Poole [4]. In this paper we will show that these coefficient problems are equivalent to extremal problems for the coefficients in the power series expansion of functions in S(a), (a < 0). This alternative approach leads to a substantial generalization. We will also prove several theorems of the type commonly called "distortion theorems" for functions in the classes S(a). One such result is the: THEOREM. Let f(z) E S(a), -oo < a < 1. Then for each natural number n there is a point z = eio on the unit circle such that

with coefficients p;,(x) belonging to V, the class of complex-valued (Lebesgue) integrable functions of the real variable x on the compact interval ab: a ?1 the subclasses of functions f(x) of Ck, 2fk and fk;2 such that f(a)(a)= 0=f(a)(b), (a =O, 1, * * *, k-1), will be denoted by 0, W', and Wk;2, respectively. In order to avoid supplementary comments at various places in the paper, it will be understood that the symbols %t0 and W' designate the class ?, the symbols 9o;2 and 2o;2 denote the class V2, and that GS designates the class (0 of functions continuous on ab. As is customary, functions fi(x), f2(x) that are equal a.e. (almost everywhere) are considered as equal, and we write fi =f2; correspondingly, for subsets Z,a C ?, (a =1, 2), we write Z1 = T2 if for each w(x) )a (a = 1, 2), there is a u(x)EEO,, (fla), such that u=w. If La(y)= Z=OpA,.(X)y(A), (a=1, 2), then Li(y) L2(y) signifies P,1 = P,,2, (4 = 0, 1, , n). Finally, if u = u(x), v==v(x) and uv belong to V the symbol (u, v) is used for fvu~dx; in particular, (u, v) exists if u, vGC2. As pueV, (,u=O, 1, * , n), if yCE(5 then L(y)C?. The operator To is defined to have domain Gn and value Toy =L(y) for y Ct . If D denotes the totality of functions z(x)C& with 2(x)pu(x)CV, (,u=O, 1, , n), and for which there exists a correspondingfz-f=(x) C? such that (L(y), z) = (y, fz) for all yGO, then the operator To with domain O* and value T*z=f, is termed the adjoint of To. In particular, if pp6u, (b=O, , *n), and pn(x) X0 on ab, then classical results provide the conclusion that T*=Sb1 and for zC2f, the value of To*z is given by the Lagrange adjoint z,=0 (-1)y(pnz)(#). A very

1. The presenit paper is devoted first to the proof of a theorem fundamental in the construction of imprimnitive linear homogeneous groups t in a given niumber of variables. Then, by means of this and earlier theorems given by the author on the subject of linear groups, t JORDAN'S theorem, j to the effect that the order of a linear homogeneous group G in n variables is of the form Xf, where f is the order of an abelian self-conjugate subgroup of G, and X is less than a fixed number depending onlv upon n, is proved for imprimitive groups, a number being found that X must divide. Finally, the principal imprimitive collineation-groups in 4 variables are found and their generating substitutions given. THEOREM. Either an imprinzitive linear homogeneous group G can be written in monomial tfrm, ? or the n variables of the group can be so selected that they fall into k sets qf imiprinit'ivity of m variables each (n = kin), permuted according to a permutation-group K in k letters, which group is transitive (in the sense of transitivity of permutation-groups). The subgroup ( G') Of G, corresponding to the subgroup (K') of K which leaves one letter unchanged, is primitive (in the sense used in linear homogeneous groups) in the m variables of the set corresponding to the letter that K' leaves unchanged. In order that G may be transitive (as a linear homogeneous group, i. e., " irreducible"), it is plainly necessary that its sets of imprimitivity contain the same number of variables, and that the permnutation-group K, permuting these sets, is transitive (as a permutation-group). We shall prove that, if the

An abelian group may be defined by the property that, in an automorphism of the group, more than three fourths its operators may be placed in a one to one correspondence with their inverses.t It may be of interest to know the groups possessing the property that five eighths or more of the operators may be inade to correspond to their inverses. The principal object of this paper, however, is to establish the following elementary theorem (I) and to illustrate the use that may be made of it in certain problems. THEOREM I. A group that has two invariant subgroulps with nothing in common, but the identity can be set up as a nlzutiple isornorphisn?, between two groups of lower order. Let a group ( G ) of order k, k x have the two invariant subgroups K1 and K, of order k1 and k2 respectively. If K, and K2 have only the identity in comiimon, every operator of K, is commutative with every operator of K2. It may be assumed that G is not merely the direct product of K, ancl K2. Let 1, r2 r31 ** , be the operators of K-1 and 1 s2, s, ..*, those of K2. Now

If a grotup (G) of order 1pcontains only one subgroup of order p, a > 0, it is known to be cyclic unless both p = 2 and c = 1.4 In this special case there are two possible groups whenever m> 2. The number of cyclic subgroups of order pa in G is divisible by p whenever G is non-cyclic and p> 2. I In the present paper we shall consider the possible types of G when it is assumed that there are just p cyclic subgrouips of order pa in G. That is, we shall consider the totality of groups of order prn which satisfy the condition that each group contains exactly p cyclic subgroups of order pa. It is evident that

Professor L. E. Dickson in his paper A fundamenttal system of invariants of the gener al miodular linear group wvith a solttion of the forvi problemt' gave a fundamental system of invariants for the group of all linear homogeneous transformations in n variables with coefficients in the Galois field of order pf, denoted by GF[pn]. Dr. J. S. Turner extended the results for the biniary group in GF[p] to the binary group H witlh coefficients ranging over integers modulo p2 and the determinant of transformation congruent to unity modulo p2, This thesis deals with the invariants of the linear homogeneous group in n variables with the coefficients ranging over integers modulo pk, and the determinant of transformation congruent to unity modulo p. The writer avails himself of this occasion to express his gratitude to Professor Dickson for the conduct of this research and for suggesting both the problem and the group-theoretic principles applied in the followinig pages.

The groups in which every two conjugate operators are cominutative have recently been considered by BURNSIDE.t In the first section of the present paper, the converse limitation is imposed on a group of operations. It is assumed that every two commutative operations are conjugate, provided neither is identity, t and the groups which are possible under this hypothesis are determined. It results that the group of order 2 ? and the symmetric group of order 6 are the only grou.ps which have the property in question. In sections 2-6, somewhat similar but smaller limitations are imposed on the group. The condition is imposed in sections 2-5 that every two commutative operations of the same order are conjugate, and in section 6 that every two commutative operations (identity excluded) are so related that each of them is conjugate to some power of the other. Some of the chief properties of the groups which are possible under these limitations are derived. The sections 7-8 deal with problems closely related to the precedilng. If it is assumed that a certain simple relation exists between the number X of complete sets of conjugate operations, and the number n of distinct prime factors in the order of the group, certain commutative operations are conjugate. Much use is made of this fact in showing what groups are possible under the given hypothesis. The symbol Ga (S1= 1), S2, S3, * , * g will be used to represent the group of order g under consideration, and p1, P2, * p Pn to represent distinct primes in ascending order of magnitude so that gp a1pj2. pa-.

It may happen that the function F is of such a character that when the 5-symbols z, x are given in (1), an c5-symbol y is uniquely determined, and when the25-symbols y, z are given in (1), an25-symbol x is uniquely determined. In this case we may associate with the function F(x, y) two other one-valued functions y = G (z, x), x = H (y, z) defined over the collection (B. These functions are called the first and second inverses of the function F. The primary object of this paper is to state the restrictions which must be imposed upon F in order that one of its inverses may define with (E an abstract group. It is convenient, in developing the properties of the system { ( ;o} consisting of the i-symbols, F, and one or more postulates, to replace (1) by the notationt z = xoy.

Let G and G' be finite groups with isomorphic holomorphs. It has been shown that if G and G' are abelian, then G and G' are isomorphic(1). The object of this paper is to show that if G is abelian, then G' is also abelian and hence isomorphic to G. Thus if G is not isomorphic to G', then neither G nor G' is abelian(2). In [5] it was pointed out that two finite groups with isomorphic holomorphs are not necessarily isomorphic. In fact if n ?3, the dihedral and dicyclic groups of order 4n have isomorphic holomorphs. 1. Definitions. Let G be a group and let A be its group of automorphisms. The set H of all pairs (g, u), gCG, uCA, forms a group under the composition

The congruence n_ x2(mod k) is possible, and n is therefore a quadratic residue of k, when n is a quadratic residue of each prime factor of k, so that in order to determilne the possibility of the congruence in all cases we must be able to determine its possibility when k is any prime number. The case k 2 is simple, but when k is an odd prime the problem presents some difficulties, and it has perhaps received more attention than any other in the theory of numbers. LEGENDRE introduced the symbol (n/k) = i 1 E= ny(k-) (mod k), the sign being + or as n is or is not a quadratic residue of the prime number k, and since his time the problem has consisted in determining the sign of (n/k) for any given values of n and k, n being prime to the odd prime k. The method of evaluation, or algorithm, of LEGENDRE, improved by JACOBI, is still the standard solution. It requires the use of the law of quadratic reciprocity formulated by LEGENDRE, though perceived earlier by EULER: theoremafundamentale, as it was called by GAUSS, who first supplied for it a satisfactory demonstration. The derivation of this law has attracted uniusual attention from many mathematicians, eight demonstrations having been prodcuced by GAUSS alone. The chief improvement since the time of JACOBI consists in an observation made independently by SCHERING and KRONECKER4 namely, that "4 GAUSS'S characteristic," ,, is available for the proof of the law of reciprocity when k is not prime. The definition (n/k) (1)', employed by TANNERY in his proof of the usual algorithm, is one of two employed in the present paper, and is herein extended and applied to wider purposes, with only the slightest reference to the law of reciprocity. I find great advantage in substituting for the symbol ,t the broader symbol , (n, k), so as to be able to discuss the function , for different values of n and k, and thereby to develop relations of the fulnctions ,t(n, k) im

It is known that the holomorph (K) of a cyclic group (G) is a complete group and that its commutator subgroup is G whenever the order (g) of G is odd. WAThen g > 2 is even, the commutator subgroup of K is the subgroup of G whose order is g/2, and K is never complete.t The main object of this paper is to determine additional useful properties of Kwhose subgroups are of such fundamental importance. In particular, we shall determine the orders of all the operators of Kand some of the properties of its group of isomnorphisms when g is even. It will be observed that the generalized FERMAT'S theorem follows directly from some of the properties of the group of isomorphisms of G. Let g 2a0palpa2 pa,. (r1 2, ., p being any odd prime numbers) and let Ko Xj, K2, K2 , K7 represent the holomorphs of the cyclic groups (GO, G, G .. , G,) of orders 2ao,plal ,pa, ,pam respectively. As K is evidently the direct product of these holomorphs j the orders of all the operators of K can be directly obtained from the orders of the operators in these holomorphs. We shall first consider the operators of Ko (6aO > 2 ) whose order is known to be 22ao1 The group of isomorphisms (I,,) of G. is known to be the direct product of an operator of order two and a cyclic group which may be so chosen that it is composed of all the operators of ]o which transform ? an operator of order 4 in G into itself. ? The orders of the two independent generators (S, 82) of ]o are therefore 2ao-2 and 2 respectively.** We shall first determine the orders of all the

then it is known, MacNeish [16] and Mann [17] that there exists a set of at least n(v) mutually orthogonal Latin squares (m.o.l.s.) of order v. It seemed plausible that n(v) is also the maximum possible number of m.o.l.s. of order v. This would have implied the correctness of Euler's [13] conjecture about the nonexistence of two orthogonal Latin squares of order v when v =2 (mod 4), since n(v) -1 in this case. If we denote by N(v) the maximum possible number of mutually orthogonal Latin squares of order v, then Parker [18 ] showed that in certain cases N(v) >n(v) by proving that if there exists a balanced incomplete block (BIB) design with v treatments, X = 1, and block size k which is a prime power then N(v) > k -2. This result cast a doubt on the validity of Euler's conjecture. In this paper we generalize Parker's method of constructing m.o.l.s., by showing that a very general class of designs, which we have called pairwise balanced designs of index unity, can be used for the construction of sets of m.o.l.s. Various applications of this method have been made. In particular we show that Euler's conjecture is false for an infinity of values of v, including all values of the form 36m+22. We also provide a table for all values of v n(v). The smallest number for which we have been able to demonstrate the falsity of Euler's conjecture is 22. Two orthogonal squares of order 22 constructed by our method are given in the Appendix. Several attempts, necessarily erroneous, have been made in the past to prove Euler's conjecture, e.g., Peterson [19], Wernicke [20] and MacNeish [16]. Levi [14, p. 14] points

found in the writings of Jacobson [9], Becker [2], Motzkin [11], and Bourbaki [3; 4]. This paper will be concerned with a natural generalization of Cayley's problem, and will show that the solution to the generalized problem contains all of the combinatorial information needed to establish the well known formula of Lagrange for the reversion of power series. To describe the problem, we consider expressions which are built from operator symbols and argument symbols, using a prefix notation for operators. Weights are assigned to the symbols in an expression, an argument symbol having the weight 0 and an n-ary operator symbol having the weight n. Expressions are of various types, the type of an expression depending only on the weights of the symbols in it and on the order in which they appear. The expression (a+b) +c, for example, is written + +abc and is of the type 22000, while a+(b+c) is written +a+bc and is of the type 20200. The expression F(G(x, H(y, z), t), K(u)) is of the type 230200010. Following P. C. Rosenbloom [13], we call those finite sequences of natural numbers which designate the types of expressions "words." Definitions and some special properties of these sequences are stated in ?2.

Every group G determines two important structural invariants, namely its central C(G) and its central quotient group Q(G) =G/C(G). Concerning these two invariants the following two problems seem to be most elementary. If A and B are two groups, to find necessary and sufficient conditions for the existence of a group G such that A and C(G), B and Q(G) are isomorphic (existence problem) and to find necessary and sufficient conditions for the existence of an isomorphism between any two groups G' and G" such that the groups A, C(G') and C(G") as well as the groups B, Q(G') and Q(G") are isomorphic (uniqueness problem). This paper presents a solution of the existence problem under the hypothesis that B (the presumptive central quotient group) is a direct product of (a finite or infinite number of finite or infinite) cyclic groups whereas a solution of the uniqueness problem is given only under the hypothesis that B is an abelian group with a finite number of generators. There is hardly any previous work concerning these problems. Only those abelian groups with a finite number of generators which are central quotient groups of suitable groups have been characterized before.t 1. Before enunciating our principal results (in ?2) some notation and concepts concerning abelian groups will have to be recorded for future reference. If G is any abelian group, then the composition of the elements in G is denoted as addition: x+y. If n is any positive integer, then nG consists of all the elements nx for x in G, and Gn consists of all the elements x in G such that nx = 0. F(G) is the subgroup of all the elements of finite order in G, that is, the join of the groups G, and F(G, p) is the subgroup of all those elements in F(G) whose order is a power of the prime number p; in other words, F(G, p) is the join of all the groups Gpn. It is well known that F(G) is the direct sum of the groups F(G, p). A set B of elements in G is termed independent, if all the elements in B are non-zero, if none of the elements in B is a multiple of another element in B, and if the group, generated by the elements in B, is the direct sum of the cyclic groups which are generated by the elements in B. If G is generated by

Let G be a group. An element g is c.d. in G if and only if given any element h of G, either it is conjugate to h or there is a homomorphism e from G onto a finite group such that e(g) is not conjugate to e(h). Following A. Mostowski, a group is conjugacy separable or c.s. if and only if every element of the group is c.d. Let F be a Fuchsian group, i.e. let F be presented as F = (S1. Sn, ai ... X a2o bi,. .., bi; Sell=* ** Sen = S1 ... Sal ... a2a ' 1 ... a-lb, ... bt = 1). In this paper, we show that every element of infinite order in Fis c.d. and if to 0 or roO, Fis c.s.

Most physical events can be resolved, in theory at least, into a set of independent components in such a way that, when the result of each component event is known, the result of the principal event is fully determined. If the question of order does not enter into the determination so that, without changing the situation, the component events may be thought of as occurring simultaneously, the relation between the component events and the principal event can be formulated analytically in the "Verbindung" operation of von Mises. However, if the order of the component events is significant-as, for example, when the principal event is a set of tennis, the outcome depending to some extent on the order of gains and losses in a series of individual games-von Mises points out that his methods are not applicable.t It is the purpose of this paper to develop a set of transformations capable of dealing with such problems. There will be associated with each transformation of this set a set of admissible numbers having the power of the continuum for which the property of admissibility is invariant under the given transformation. Furthermore, every denumerable subset of such transformations will be shown to have a similar property. The meaning of admissibility may be explained as follows. Assume that a one-to-one correspondence has been established between the set of all positive integers X and the set of all sets of integers n, rl, r2, , rk, such that O