Showing papers in "Canadian Journal of Mathematics in 1971"

Transitive Orientation of Graphs and Identification of Permutation Graphs

Abstract: The graphs considered in this paper are assumed to be finite, with no edge joining a vertex to itself and with no two distinct edges joining the same pair of vertices. An undirected graph will be denoted by G or (V, E), where V is the set of vertices and E is the set of edges. An edge joining the vertices i,j ∊ V will be denoted by the unordered pair (i,j). An orientation of G = (V, E) is an assignment of a unique direction i → j or j → i to every edge (i,j) ∊ E. The resulting directed image of G will be denoted by G→ or (V, E→), where E→ is now a set of ordered pairs E→ = {[i,j]| (i,j) ∊ E and i → j}. Notice the difference in notation (brackets versus parentheses) for ordered and unordered pairs.

Sphere packings and error-correcting codes

Abstract: Error-correcting codes are used in several constructions for packings of equal spheres in ^-dimensional Euclidean spaces En. These include a systematic derivation of many of the best sphere packings known, and construction of new packings in dimensions 9-15, 36, 40, 48, 60, and 2m for m g 6. Most of the new packings are nonlattice packings. These new packings increase the previously greatest known numbers of spheres which one sphere may touch, and, except in dimensions 9, 12, 14, 15, they include denser packings than any previously known. The density A of the packings in En for n = 2m satisfies log A ~ — log log n as n —* oo. 1.1. Introduction. In this paper we make systematic use of error-correct ing codes to obtain sphere packings in En, including several of the densest packings known and several new packings. By use of cross-section s we then obtain packings in spaces of lower dimension, and by building up packings by layers we obtain packings in spaces of higher dimension. Collectively, these include all of the densest packings known, and further new packings are also con­ structed. Part 1 of the paper is devoted to groundwork for the constructions. § 1.2 introduces sphere packings, and §§ 1.3-1.8 survey the error-correcting code theory used in the later Parts. Part 2 describes and exploits Construction A, which is of main value in up to 15 dimensions. Part 3 describes Construction B% of main value in 16-24 dimensions. Part 4 digresses to deal with packings built up from layers, while Part 5 gives some special constructions for dimen­ sions 36, 40, 48 and 60. Part 6 deals with Construction C, applicable to dimensions n = 2m and giving new denser packings for m ^ 6. We conclude with tables summarizing the results. Table I, for all n S 24, supersedes the tables of [18; 19], and Table II gives results for selected n > 24. The tables may be used as an index giving references to the sections of the paper in which the packings are discussed. Partial summaries of this work have appeared in [22; 23]. General references for sphere packing are [18; 19; 31] and for coding theory [4; 25].

Subgroups of HNN Groups and Groups with one Defining Relation

Abstract: HNN groups have appeared in several papers, e.g., [3; 4; 5; 6; 8]. In this paper we use the results in [6] to obtain a structure theorem for the subgroups of an HNN group and give several applications. We shall use the terminology and notation of [6]. In particular, if K is a group and {φ i } is a collection of isomorphisms of subgroups {L i} into K, then we call the group 1 the HNN group with base K, associated subgroups { Li,φi (Li )} and free part the group generated by t1, t2, …. (We usually denote φi (Li ) by Mi or L –i.) The notion of a tree product as defined in [6] will also be needed.

Normed Linear Spaces that are Uniformly Convex in Every Direction

Abstract: The concept of uniform convexity in a normed linear space is based on the geometric condition that if two members of the unit ball are far apart, then their midpoint is well inside the unit ball. We consider here a generalization of this concept whose geometric significance is that the collection of all chords of the unit ball that are parallel to a fixed direction and whose lengths are bounded below by a positive number has the property that the midpoints of the chords lie uniformly deep inside the unit ball. This notion, called uniform convexity in every direction (UCED), was first used by A. L. Garkavi [5; 6] to characterize normed linear spaces for which every bounded subset has at most one Cebyŝev center. We discuss questions of renorming spaces so as to be UCED and forming products of spaces that are uniformly convex in every direction.

The free product of two groups with a malnormal amalgamated subgroup

Abstract: In [1], B. Baumslag defined a subgroup U of a group G to be malnormal in G if gug –1 ∈ U, 1 ≠ u ∈ U, implies that g ∈ U. Baumslag considered the class of amalgamated products (A * B; U) in which U is malnormal in both A and B. These amalgamated products play an important role in the investigations of B. B. Newman [13] of groups with one defining relation having torsion. In this paper, we shall be concerned primarily with a generalization of this class. Let U be a subgroup of a group G and let u ∈ U. Then the extended normalizer EG(u, U) of u relative to U in G is defined by if u ≠ 1, and by EG(u, U) = U if u = 1.

The Dimensions of Irreducible Tensor Representations of the Orthogonal and Symplectic Groups

Abstract: It is well known [13] that the irreducible tensor representations (IRs) of the unitary, orthogonal, and symplectic groups in an n-dimensional space may be specified by means of Young tableaux associated with partitions (σ)s = (σ 1, σ 2, …, σp ) with σ 1 + σ 2 + … + σp = s. Formulae for the dimensions of the corresponding representations have been established [1; 8; 9; 13] in terms of the row lengths of these tableaux. It has been shown [12] for the unitary group, U(n), that the formula may be written as a quotient whose numerator is a polynomial in n containing s factors, and whose denominator is a number independent of n, which likewise may be expressed as a product of s factors. This formula is valid for all n.

Finite linear groups of degree six

Abstract: 1. I n t r o d u c t i o n . In this paper we classify finite groups G with a faithful, quasiprimitive (see Nota t ion) , unimodular representation X with character x of degree six over the complex number field. There are three gaps in the proof which are filled in by [16; 17]. These gaps concern existence and uniqueness of simple, projective, complex linear groups of order 604800, |LF(3 , 4) | , and |PSL4(3) | . By [19], X is a tensor product of a 2-dimensional and a 3-dimensional group, or a subgroup thereof, or X corresponds to a projective representat ion of a simple group, possibly extended by some automorphisms. T h e tensor product case is discussed in section 10. Otherwise, we assume t h a t G/Z(G) is simple. W e discuss which automorphisms of G/Z(G) extend the representat ion X ( tha t is, lift to the central extension G and fix the character corresponding to X) jus t after we find X(G). All cases where the simple groups G/Z(G) have an irreducible projective complex representation of degree 2, 3, 4, 5, or 7 are discussed in section 11, where we use the corresponding classifications of Blichfeldt, Brauer, and Wales. Otherwise, we assume t h a t no such projective representations exist. By section 6, we are allowed to assume t h a t no prime greater than 7 divides |G|. By [18], if 7 divides \\G\\, then G has a nontrivial , normal 7-group, or X is reducible, or G has a subgroup of index 2, all contrary to the simplicity of G/Z{G). T h e case where 7 does not divide \\G/Z(G)\\ is discussed in section 5. Otherwise, we assume t h a t 7 divides \\G/Z(G)\\ to the first power. As we principally use the degree equation of [4] for the prime 7, we break this case up into subcases for [N(S^ : C ^ ) ] . These cases are treated in sections 7 and 8.

Going Down in Polynomial Rings

Abstract: In this paper, R ⊂ T will be commutative domains having a common identity. Definition. Suppose that R is a subdomain of T. (i) If P is a prime ideal of R and Q is a prime ideal of T, we say that Q lies over P if Q ∩ R = P. (ii) If every prime of R has a prime of T lying over it, we say that R ⊂ T has lying over. (iii) If there is a unique prime of T lying over P in R, we say that P is unibranched in T. (iv) If every prime of R is unibranched in T we say that R ⊂ T is unibranched.

The Minimal Prime Spectrum of a Commutative Ring

Abstract: We call a topological space X minspectral if it is homeomorphic to the space of minimal prime ideals of a commutative ring A in the usual (hull-kernel or Zariski) topology (see [2, p. 111]). Note that if A has an identity, is a subspace of Spec A (as defined in [1, p. 124]). It is well known that a minspectral space is Hausdorff and has a clopen basis (and hence is completely regular). We give here a topological characterization of the minspectral spaces, and we show that all minspectral spaces can actually be obtained from rings with identity and that open (but not closed) subspaces of minspectral spaces are minspectral (Theorem 1, Proposition 5).

Maximal abelian subgroups of the symmetric groups

Abstract: 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 Erdos and Turan deal with properties of the set of elements of Sn. The present work arose out of a conversation with Professor Turan 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?

On unitary polarities of finite projective planes

Abstract: 1. In t roduc t ion . A unitary polarity of a finite projective plane 8P of order q is a polarity 0 having q + 1 absolute points and such that each nonabsolute line contains precisely q + 1 absolute points. Let G{6) be the group of collineations of SP centralizing 6. In [15] and [16], A. Hoffer considered restrictions on G (6) which force SP to be desarguesian. The present paper is a continuation of Hoffer's work. The following are our main results.

A sufficient condition that an operator algebra be self-adjoint

Abstract: It is well-known, and easily verified, that each of the following assertions implies the preceding ones. (i) Every operator has a non-trivial invariant subspace. (ii) Every commutative operator algebra has a non-trivial invariant subspace, (iii) Every operator other than a multiple of the identity has a non-trivial hyperinvariant subspace. (iv) The only transitive operator algebra on is Note. Operator means bounded linear operator on a complex Hilbert space , operator algebra means weakly closed algebra of operators containing the identity, subspace means closed linear manifold, a non-trivial subspace is a subspace other than {0} and , a. hyperinvariant subspace for A is a subspace invariant under every operator which commutes with A, a transitive operator algebra is one without any non-trivial invariant subspaces and denotes the algebra of all operators on .

The Lattice of Equational Classes of Commutative Semigroups

Abstract: There has been some interest lately in equational classes of commutative semigroups (see, for example, [2; 4; 7; 8]). The atoms of the lattice of equational classes of commutative semigroups have been known for some time [5]. Perkins [6] has shown that each equational class of commutative semigroups is finitely based. Recently, Schwabauer [7; 8] proved that the lattice is not modular, and described a distributive sublattice of the lattice. The present paper describes a “skeleton” sublattice of the lattice, which is isomorphic to A × N + with a unit adjoined, where A is the lattice of pairs (r, s) of non-negative integers with r ≦ s and s ≧ 1, ordered component-wise, and N + is the natural numbers with division.

A Note on Supersoluble Groups

Abstract: Let H be a subgroup of a group G (all groups considered throughout this article are finite); then H will be called primitive if the subgroup is distinct from H. Such subgroups, which are also called meet-irreducible, arise naturally in connection with minimal permutation representations of groups and in other contexts; for example, every subgroup of a group G can be written as an intersection of primitive subgroups of G, and the set of all primitive subgroups of G is characterized by its minimality with respect to this property. While maximal subgroups are always primitive, most groups contain non-maximal subgroups which are primitive (see remark at end of article). Note that a subgroup H of an abelian group G is primitive if, and only if, G/H is cyclic of prime-power order.

On branched coverings of $Ssp{3}$

Abstract: In [3] Fox studied a certain class of irregular coverings of S3 branched along some knot or link which turned out to be homotopy spheres. By a simple geometric construction, it is shown in this paper that these homotopy spheres are just 3-spheres, provided that the group of the knot or link k in question cannot be generated by a number of Wirtinger generators smaller than the minimal number of bridges of this knot or link. The knots and links with two bridges provide examples for such coverings. In the covering sphere there is a link covering k. With the help of braid automorphisms, can be determined. Figure 3 shows a link in a 5-sheeted covering over k = 41. Links over 31 and 61 in 3-sheeted coverings were determined by Kinoshita [5] by a different method. The method used here is applicable to these cases and confirms his results.

On the partially ordered set of prime ideals of a distributive lattice

Abstract: For a distributive lattice L, let denote the poset of all prime ideals of L together with ∅ and L. This paper is concerned with the following type of problem. Given a class of distributive lattices, characterize all posets P for which for some . Such a poset P will be called representable over . For example, if is the class of all relatively complemented distributive lattices, then P is representable over if and only if P is a totally unordered poset with 0, 1 adjoined. One of our main results is a complete characterization of those posets P which are representable over the class of distributive lattices which are generated by their meet irreducible elements. The problem of determining which posets P are representable over the class of all distributive lattices appears to be very difficult.