Marshall Hall

Distinct representatives of subsets

If we make this assumption it follows that 2JÎ annuls dA/dyir, where r is the order of A in yx. Let s be the order of A in y%. We form the resultant R of A and dA/dyir, considered as algebraic polynomials in y%8. Since A is irreducible, and cannot be a factor of dA/dyir, R is a nonzero polynomial, free of y%9, which is annulled by 9K. Since R is of lower efiective order than A in y%, 5DÎ must be an essential singular manifold of A relative to y^ The proof is now complete.

Codes and designs

In addition to its own very considerable merits, coding theory has recently become a valuable tool for investigating block designs. It is hoped that coding theory will be as useful in the study of designs as representation theory has been in the study of groups. MacWilliams et al. [5] used the binary code of the plane of order 10 to investigate its properties. A recent paper by Anstee et al. [I] showed that a plane of order 10 cannot have a collineation of order 5. It has been shown by Z. Janko (personal communication) that there is no collineation of order 3. Together with earlier results it now follows that a plane of order 10 can have only the identical coilineation. This leaves the code as the main tool for investigating planes of order 10 if any exist, or for showing non-existence of such a plane. This paper undertakes a general study of the application of coding theory to designs. Section 2 gives the standard facts on block designs (partially balanced incomplete block designs or PBIB for short) with parameters u, b,

A new construction for Hadamard matrices

An Hadamard matrix H is a. square matrix of ones and minus ones whose row (and hence column) vectors are orthogonal. The order n of an Hadamard matrix is necessarily 1, 2 or At with / = 1, 2, 3, • • I t has been conjectured that this condition (n = 1, 2 or At) also insures the existence of an Hadamard matrix. Constructions have been given for particular values of n and even for various infinite classes of values. While other constructions exist, those given by [ l ] [7] exhaust the previously known values of n. This paper gives a new construction which yields, among others, the previously unknown value « = 156, leaving only two undecided values of « = 42^200 (these are 116 and 188). An Hadamard matrix is said to be of the Williamson type if it has the structure imposed by Williamson [6], that is

Discovery of an Hadamard matrix of order 92

An Hadamard matrix H is an n by n matrix all of whose entries are + 1 or — 1 which satisfies HH = n J, H being the transpose of H. The order n is necessarily 1, 2 or 42, with t a positive integer. R. E. A. C. Paley [3] gave construction methods for various infinite classes of Hadamard matrices, chiefly using properties of quadratic residues in finite fields. These constructions cover all values of 4 ^ 2 0 0 , except 4/ = 92, 116, 156, 172, 184, 188. Further constructions have been given by J. Williamson [5; 6] , A. Brauer [ l ] , M. Hall [2] and R. Stanton and D. Sprott [4]. Williamsons first paper gave an Hadamard matrix of order 172, incorporating a special automorphism of order 3. The same method may be applied to 92, 116, 156, and 188, but Williamson did not do so, principally because of the amount of computation involved. Williamsons method has been applied to 4^ = 92 using the IBM 7090 at the Jet Propulsion Laboratory. The matrix H has the form

Groups Generated by a Class of Elements of Order 3

Abstract : The Conway group which is the group of automorphisms of the 24 dimensional Leech lattice is generated by a class of elements of order 3 with the property that any two of them either commute or generate SL2(3), SL2(5) or the alternating groups A4, A5 which are isomorphic to SL2(3) and SL2(5) modulo a center of order 2. Such a class of order 3 is also a special case of John Thompsons Quadratic pairs for the prime 3. The paper is restricted to elements of order 3 in which any two either commute, generate A4 or SL2(3). It is possible to describe these groups completely. (Author)

Designs derived from permutation groups

Abstract Let G be a transitive permutation group on a set Ω of v points {1, 2, …, v}. Let H be an intransitive subgroup of G and let Δ a set of k points where Δ consists of complete orbits of H. Then the images Δx of Δ under permutations x of Δ have been shown by the first author to be a partially balanced block design D with G as a group of automorphisms. Under certain circumstances D is a balanced incomplete block design. Here a representation of the simple group PSL3(4) of order 20,160 on 56 letters leads to a new symmetric block design with parameters v=56, k-11, λ=2. A representation of the simple group of order 25,920 as U4(4) on 45 isotropic points gives a symmetric design with v=45, k=12, λ=3. One representation of U4(4) on 40 points, gives the design of planes in PG(3, 3) and exhibits the isomorphism of this group to the symplectic group S4(3).

Planes of order 10 do not have a collineation of order 5

The existence of a projective plane of order 10 remains in doubt. If one does exist it may have only the identity collineation. D. R. Hughes [I, 2] showed that for a plane of order n where y1 = 2 (mod 4) and IZ > 2 the collineation group is of odd order. He also showed that for a plane of order IO the only primes dividing the order of the collineation group could be 3, 5, or 11, that for order 3 there would be 3 or 9 fixed points (also lines) and for 5 exactly one fixed point and one fixed line. Whitesides [9] has eliminated the possibility of a collineation of order Il. She has also [lo] eliminated orders 9,25, and 15, so that the only remaining possible orders are 1, 3, or 5. The main result of this paper is to eliminate the order 5.

Incidence axioms for affine geometry

In terms of incidence alone it is possible to define an affine plane, as Artin does [I], by calling lines parallel if they do not intersect, and basing the definition on the Euclidean axiom that there is a unique parallel to a line through a point not on the line. In higher dimensions we can define afine geometry by deleting the points and lines of a hyperplane from a projective geometry, using the axioms of Veblen and Young [4]. It is an easy exercise to show that the Artin approach and that of Veblen and Young agree in the definition of an afIine plane. Sut in higher dimensions it is not clear how an afline geometry can be defined directly so that it can be shown to arise from a projective geometry by deleting the points and lines of a hyperplane. This paper gives a set of axioms which have this property. We must define parallelism in such a way that nonintersecting lines in a plane are parallel. But the real surprise is that this is not enough. In Section 5 an example is given of a geometry in which every plane is an afhne plane, but the geometry as a whole is not an afline geometry. If we think of the embedding of an affine geometry into a projective geometry, lines are parallel if and only if they pass through the same projective point at infinity. In particular, the abstract property of parallelism between lines must be transitive, and three parallel lines need not lie in a plane. Thus the axiom of transitivity of parallelism (Axiom A4 in Section 2) is a three-dimensional axiom. The axioms are given in Section 2. Affine planes are defined in Section 3, and their main properties are developed there. The issue as to whether or not the plane is Desarguesian does not arise. In Section 4, the ideal (infinite) points and lines are defined and adjoined to the affine geometry. It is shown that the resulting geometry is a projective geometry. From the treatment here, it follows that if we have an incidence geometry, consisting of points and certain distinguished subsets of points which we call

A search for simple groups of order less than one million

Publisher Summary This chapter describes the search for simple groups of order less than one million. In 1900, L. E. Dickson listed 53 known simple groups of composite order less than one million. Three more groups have been added to this list since that time. A group of order 29,120 was discovered by M. Suzuki in 1960, the first of an infinite class, and one of order 175,560 was discovered by Z. Janko in 1965, which appears to be isolated. Z. Janko announced that a simple group with certain properties would have order 604,800 and have a specific character table. The chapter describes the construction of a simple group of order 604,800. The notation for the classical simple groups presented in this chapter is essentially that used in Artin. GF(q) denotes the finite field with q elements where q = pr, p being a prime. The chapter also discusses the role of Brauer theory of modular characters in this search, particularly the theory for groups whose order is divisible by exactly the first power of a prime.

On the number of Sylow subgroups in a finite group

If the order g of a finite group G is divisible by pr but no higher power of the prime p, then the classical theorems of Sylow [A assert that there are subgroups of order p+‘, the Sylow p-subgroups, forming a single conjugate class in G, and that the number of these, 1z9 is of the form ng = 1 + kp for some integer k >, 0. If G is solvable, it was shown by P. Hall [5] that n, is a product of factors of the form qt, where q is a prime and qt E 1 (mod p). For a simple group X the number sp of Sylow p-subgroups need not be of this form. It is shown in this paper (Theorem 2.2) that the number n, of Sylow p-subgroups in any finite group G is a product of factors of the form (I) sp , where there is a simple group X with s, Sylow p-subgroups and (2) a power qt of a prime q, where qt = 1 (modp), and that an arbitrary product of factors of these two kinds is the n, of some finite group. Thus the rz=‘s form a semigroup. A quotient of two 11,‘s which is an integer need not be an np, , since LF(2.7) has eight Sylow 7-subrgoups and A, has 120 Sylow 7-subgroups, but no group has 15 Sylow 7-subgroups, a fact proved in Theorem 3.1. Any odd number may be the number of Sylow 2-subgroups in a finite group. But for every prime p > 2, not every integer it = 1 (mod p) is the n, of a finite group. In particular Theorem 3.2 shows that there is no finite group with n, = 22, n6 = 21, or n, = 1 + 3p for p > 7. As with so many questions, the mysterious part as to the possible values for a9 lies in the study of the finite simple groups.