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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

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

Several recent papers have been concerned with a group G, of order 28.36.5.7, which can be represented as a collineation group in five dimensions. This collineation group is generated by harmonic inversions (projections) which leave fixed a point (the vertex) and a prime (said to be conjugate to the vertex). There are 126 projections in G and the set of 126 vertices form a configuration which is described in a paper (Hamill (3)) in which the operations of G are expressed as products of at most six projections. The group leaves invariant six algebraically independent primals; these have been determined (Todd (6)), and the equations of the simplest are given. The simplest invariant is a sextic primal, and some properties of this have been recorded in an earlier paper (Hartley (4)).

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

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.

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

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

The paper derives an asymptotic iterated (two-fold) logarithm of the number of simple bases in Pk, as well as the maximal order of the functions prime for each precomplete class in P3.

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

NOTATION AND DEFINITIONS. A group G has property E if it can be embedded in a sfield D and property EE if every automorphism of the group G can be extended to be an automorphsim of D. For a more complete discussion see [5 ]. The result on locally finite groups depends mainly on a result of Amitsur (see [I]). A complete discussion of the following notation can be found in his paper. 7r will denote the set of all primes and ri the set of all odd primes p such that 2 has odd order mod p. Let m and r be relatively prime integers. Put s = (r -1, m), t = m/s and n = minimal integer satisfying rn= 1 mod m. Denote by Gm,r a group generated by two elements A and B satisfying Am=1, Bn=At and BAB-'= A. Denote by G., a group G which has a countable ascending tower of subgroups Hi: O < i< oo } such that G = U1 Hi and each Hi is isomorphic to Gmi,ri. T*, 0*, 1* will denote the binary tetrahedral, octahedral and icosahedral groups. Let p be a fixed prime dividing m. a= aP is the highest power of p dividing m. 7, is the minimal integer satisfying rap= 1 mod (Mp-a). AuP is the minimal integer satisfying r/,P=p'2 mod (Mp-a) for some integer ,u. bp is the minimal integer such that ppr_ 1 mod (mp-a).

In this paper the Hughes conjecture for finite pgroups is proved in the case of groups which are nilpotent of class 2p-2. If G is a group and p is a prime number then the Hughes subgroup HpiG) of G is by definition the subgroup generated by all elements of G that do not have order p. The conjecture of Hughes was that if G>HpiG)>l then HviG) must have index p in G; see [o]. Hughes showed that the conjecture is true for p = 2 and all G, and Straus and Szekeres [lO] proved it for p = 3 and all G. Hughes and Thompson [7] proved the corresponding result for all p and for G finite and nonnilpotent. The most interesting cases which remain are those in which G is a finite 7^-group with p ^ 5 ; throughout this note we shall take G to be a finite £-group, and our results will be of most significance when pit5. We may remark at this point that it is a nontrival exercise to construct a £-group GP for each prime p such that Gp>HPiGp) > 1. Though Wall [ll] has shown by means of an example that the Hughes conjecture is false for finite 5-groups and p — 5,it is nevertheless of some interest to find sufficient conditions for the conjecture to hold. Thus Zappa [12] has shown that all is well when G is nilpotent of class p. The main result of the present note is a substantial strengthening of this: Theorem. The Hughes conjecture is valid for finite p-groups of nilpotency class 2p — 2. It seems likely to us that class 2p — 2 cannot be replaced by a much greater class without falsifying the theorem. The proof of this theorem depends heavily on the main result of [8], which we state again in the interests of clarity: Theorem [8]. If \G:HPiG)\^p2, V„(G)£t(G) and y2p(G) = l, thenVPiHpiG)r\y2iG)) = l. Received by the editors February 3, 1970. AMS 1970 subject classifications. Primary 20D15, 20D25.

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 shown in this paper that if G is a finite group which is the product of two normal supersolvable subgroups of relatively prime index, then G is supersolvable. In 1953, Huppert [1] gave an example showing that, in general, the product of two normal supersolvable subgroups of a finite group need not be supersolvable. The purpose of this note is to show that if the subgroups have relatively prime index in their product, then the product is again supersolvable.2 THEOREM. Let A, B be normal supersolvable subgroups of the finite group G with G = AB. If the indices [G: A ] and [G: B ] are relatively prime, then G is supersolvable. PROOF. Assume that the theorem is false, and that G is a counterexample of minimal order. Thus we may assume that the indices of A and B in G are distinct primes p and q respectively with A/A CB cyclic of order q and B/AnB cyclic of order p. Let a(AnB) and i3(A CnB) generate these quotients and assume that a has order q and j has order pt for some s, t > 1. Let E1 be a minimal normal subgroup of A contained in ACOB. If AOB is trivial, the result follows immediately so E1 is nontrivial and has order a prime r. If E1 is normal in G, then by the minimality of G, we must have G/E1 supersolvable; and since E1 is cyclic, G itself must be supersolvable. Thus E1 has p distinct conjugates E1, E2, * * *, E.. Then each Ei< A and hence E = (E1, E2, * * *, Ep), the group generated by the Ei is an elementary abelian r-group of order

The purpose of this paper is to rephrase a conjecture about simple groups into the language of linear algebra. Let G be a group of finite order o(G). Then by rF we shall mean the group ring of G over a field of characteristic p (for instance the integers modulo p). We shall denote the radical of rF by N,. If p = 0 or p o(G), then it is known that Np=(O); and if p|o(G), Np (O). We now consider the following two assertions: (A) If G is a simple group of odd order, o(G) is a prime. (B) If G is a group of odd order o(G), then for some prime p, p[ o(G), we can find a gCG, g1, such that g-1CNp. The theorem which we propose to prove is

The present note aims to exhibit some elementary relations between wellknown methods of finding the primitive roots of a number and the properties of the cyclic group. Incidentally we arrive at a fundamental theorem relating to the primitive roots of a special class of numbers. A corollary of this theorem gives the primitive roots of all the prime numbers of the form 2p + 1, p being a prime, while it has been customary in the works on the theory of numbers to devote two theorems to the primitive roots of such prime numbers.* The note has close contact with the paper published in this JOURNAL under the title "Some Relations between Number Theory and Group Theory" and may be regarded as a continuation of this article.t It is known that the necessary and sufficient condition that a number g has primitive roots is that the cyclic group G of order g has a cyclic group of isomorphisms I. The numbers which are less than g and prime to it may be made to correspond to the operators of I, unity corresponding to the identity, in such a way that I and the group formed by these numbers, when they are combined by multiplication and the products reduced with respect to modulus g, are simply isomorphic. The orders of the operators of I are the indices of the exponents to which the corresponding. numbers belong. In particular, g -1 corresponds to the operator of order 2 and the primitive roots of g correspond to the operators of highest order in L Hence the method of finding the primitive roots of a number is equivalent to that of finding the operators of highest order in a cyclic group. One of the most instructive methods for finding all the primitive roots of g is analogous to the method known as the "Sieve of Eratosthenes" for finding