(PHD, 1981)

Abstract:

The thesis consists of three chapters. The first chapter introduces the basic notions of graph theory and defines vertex-reconstruction and edge-reconstruction problem. The second chapter and third chapter are devoted to the edge-reconstruction of bi-degreed graphs and bipartite graphs respectively.

A bi-degreed graph G is a graph with two degrees d > δ. By elementary arguments we can assume d = δ + 1 and there are at least two vertices of degree δ. Call vertices of degree d “big” vertex and degree δ “small” vertex. Define “symmetric” path of length p S_{p} to be one with both ends small vertices and all other internal vertices big vertices; define “asymmetric” path of length p A_{p} to be one with one end a small vertex and all others big vertices. If s(G) is the minimum distance between two small vertices in G, we can show that s(G) is “independent” of G (i.e. it is edge-reconstructable), and that G has at most one nonisomorphic edge-reconstruction H. From this, the concept of “forced move” posed by Dr. Swart is obvious. Using the principle of forced move (and sometimes also “forced edge” posed by Dr. Swart as well), it’s easy to derive a few interesting properties, like say G is edge-reconstructable if s(G) is even or if two S_{s(G)}’s intersect at an internal vertex, etc. Write s for s(G). When s is odd, consider the concept of s - n-chain, which is n S_{s}’s following from end to end. We can show first s - 3-chain and then s - 2-chain cannot exist. Hence S_{s}’s are disjoint. Think of S_{s}’s as “lines” in some geometry. Define two more “distance” functions s_{1} and s_{2} such that s_{1} “represents” the distance from a point to a line and s_{2} means the distance between to “skew” lines. With the aid of forced move principle again, we can at last prove every bi-degreed graph with at least four edges is edge-reconstructable.

A bipartite graph G is a graph whose vertex set V is the disjoint union of two sets v_{1} and v_{2} such that every edge joins v_{1} and v_{2}. By elementary reduction we can assume G to be connected. We define special chains inductively so that it starts at a vertex of minimum degree and always goes to a neighbor or minimum degree. Special chains will be the main tool to prove edge-reconstructability. By G’s finiteness, we note they will “terminate” somewhere, and we have three types of termination for them. Let condition A•s be that degree sequence of special chain is edge-reconstructable, condition B_{i}’s be that number of special chains is edge-reconstructable(and some more general variations); condition P’s be that the “last vertices” of two special chains be not adjacent; we can prove that all A, B_{i} and P’s should hold inductively in an interlocked way. (This is a big task). Then condition P’s can be used to prove G’s edge-reconstructability for all three types of termination. We can then prove every bipartite graph with at least four edges is edge-reconstructable.

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(PHD, 1978)

Abstract:

NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document.

Results are derived on rational solutions to […] where B is integral and A need not be square. It is shown that in general, provided a rational solution exists, one can be found in which all denominators are a power of two. More general restrictions follow from the corresponding restrictions possible on rational lattices representing integral positive definite quadratic forms of determinant one. Results due to Kneser and others are applied to show that A may be taken as integral if it has no more than seven columns, half-integral if it has no more than sixteen columns.

These results are then applied to three types of matrix completion problems, integral matrices satisfying […], partial Hadamard matrices and partial incidence matrices of symmetric block designs. It is found that rational normal completing matrices in which all denominators are powers of two are always possible in the first two cases and almost always possible in the final case.

Using a computer approach, the specific problem of showing that the last seven rows of a partial Hadamard matrix or a partial incidence matrix (with suitable parameters) can always be completed is tackled and it is shown that this is in fact the case, extending results by Marshall Hall for no more than four rows. An appendix lists the computer tabulation which is the basis of this conclusion.

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(PHD, 1974)

Abstract:

NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document.

If G is an automorphism group of a Steiner triple system which is doubly transitive on the points, then it is transitive on the blocks. It is shown that the converse is false and that all counterexamples have odd order. All Steiner triple systems which have a block-transitive but not doubly point-transitive group of automorphisms are described. They include the Euclidean geometries of odd dimension over GF(3), a class of systems first described by Netto in 1893, and another class of systems. A system in this third class has a group of automorphisms acting regularly on the blocks, and the number of points is a prime power congruent to 7 modulo 12. The number of such systems (up to isomorphism) with a prime number of points p, where […] (mod 12), is shown to be in the interval […].

The classification of block-transitive Steiner triple systems is applied to prove the following theorem: if G is a doubly transitive automorphism group of a Steiner triple system and P is a p-subgroup of G maximal subject to the condition that it fix more than three points, then the points fixed by P form a subsystem with a doubly transitive automorphism group.
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(PHD, 1969)

Abstract:

NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document.

In this thesis we study rank 4 permutation groups. A rank 4 group is a finite transitive permutation group acting on a set [Omega] such that the subgroup fixing a letter breaks up [Omega] into 4 orbits. The main tool employed in examining rank 4 groups is the use of intersection matrices, an idea introduced by Donald Higman. Intersection matrices are used to obtain relations between the lengths of the four orbits associated with a rank 4 representation and the degrees of the irreducible characters in the permutation character of the representation. It is shown that two orbits of the representation are paired if and only if two of the characters are complex conjugates of one another. All the maximal primitive rank 4 groups are determined.

Techniques are developed, using intersection matrices, to find all rank 4 representations of known finite groups. Group theoretic results about possible rank 4 groups are derived from the intersection matrices which would have to correspond to the rank 4 representation.

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(PHD, 1968)

Abstract:

Combinatorial configurations known as t-designs are studied. These are pairs ˂B, ∏˃, where each element of B is a k-subset of ∏, and each t-design occurs in exactly λ elements of B, for some fixed integers k and λ. A theory of internal structure of t-designs is developed, and it is shown that any t-design can be decomposed in a natural fashion into a sequence of “simple” subdesigns. The theory is quite similar to the analysis of a group with respect to its normal subgroups, quotient groups, and homomorphisms. The analogous concepts of normal subdesigns, quotient designs, and design homomorphisms are all defined and used.

This structure theory is then applied to the class of t-designs whose automorphism groups are transitive on sets of t points. It is shown that if G is a permutation group transitive on sets of t letters and ф is any set of letters, then images of ф under G form a t-design whose parameters may be calculated from the group G. Such groups are discussed, especially for the case t = 2, and the normal structure of such designs is considered. Theorem 2.2.12 gives necessary and sufficient conditions for a t-design to be simple, purely in terms of the automorphism group of the design. Some constructions are given.

Finally, 2-designs with k = 3 and λ = 2 are considered in detail. These designs are first considered in general, with examples illustrating some of the configurations which can arise. Then an attempt is made to classify all such designs with an automorphism group transitive on pairs of points. Many cases are eliminated of reduced to combinations of Steiner triple systems. In the remaining cases, the simple designs are determined to consist of one infinite class and one exceptional case.

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(PHD, 1967)

Abstract:

This thesis deals with the problem of how the elements from a finite field F of characteristic p are distributed among the various linear recurrent sequences with a given fixed characteristic polynomial fε F[x]. The first main result is a method of extending the so-called “classical method” for solving linear recurrences in terms of the roots of f. The main difficulty is that f might have a root θ which occurs with multiplicity exceeding p-1; this is overcome by replacing the solutions θ^{t}, tθ^{t}, t^{2}θ^{t}, …, by the solutions θ^{t}, (t_{1})θ^{t}, (t_{2})θ^{t}, …. The other main result deals with the number N of times a given element a ε F appears in a period of the sequence, and for a≠0, the result is of the form N≡0 (mod p^{ε} where ε is an integer which depends upon f, but not upon the particular sequence in question. Several applications of the main results are given.

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(PHD, 1966)

Abstract:

If *E* and *F* are saturated formations, we say that *E* is strongly contained in *F* if for any solvable group G with *E*-subgroup, E, and *F*-subgroup, F, some conjugate of E is contained in F. In this paper, we investigate the problem of finding the formations which strongly contain a fixed saturated formation *E*.

Our main results are restricted to formations, *E*, such that *E* = {G|G/F(G) ϵ*T*}, where *T* is a non-empty formation of solvable groups, and F(G) is the Fitting subgroup of G. If *T* consists only of the identity, then *E*=*N*, the class of nilpotent groups, and for any solvable group, G, the *N*-subgroups of G are the Carter subgroups of G.

We give a characterization of strong containment which depends only on the formations *E*, and *F*. From this characterization, we prove:

If *T* is a non-empty formation of solvable groups, *E* = {G|G/F(G) ϵ*T*}, and *E* is strongly contained in *F*, then

- there is a formation
*V*such that*F*= {G|G/F(G) ϵ*V*}.- If for each prime p, we assume that
*T*does not contain the class,*S*_{p’}, of all solvable p’-groups, then either*E*=*F*, or*F*contains all solvable groups.This solves the problem for the Carter subgroups.

We prove the following result to show that the hypothesis of (2) is not redundant:

If

*R*= {G|G/F(G) ϵ*S*_{r’}}, then there are infinitely many formations which strongly contain*R*.

- If for each prime p, we assume that

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(PHD, 1965)

Abstract:

A connection is shown between Konig’s Theorem on 0-1 matrices and theorems giving sufficient conditions, in terms of certain forbidden subgraphs, for a graph G to have chromatic number equal to the maximum number of vertices in any clique of G. A conjecture is proposed which would, if true, give the best possible such theorem. Three special cases of this conjecture are proved, and Konig’s Theorem is shown to be an easy corollary of any one of them.

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(PHD, 1965)

Abstract:

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(PHD, 1965)

Abstract:

In this paper, we study finite transitive permutation groups in which only the identity fixes as many as three letters, and in which the subgroup fixing a letter is self normalizing. If G is such a group, the principal results concern the case when G is simple.

In this case, H, the subgroup fixing a letter, is a Frobenius group, MQ, with kernel M and complement Q. If |H| is even we show that either G is doubly transitive or permutation isomorphic to the representation of A[subscript 5] on ten letters.

If |H| is odd we prove that Q is cyclic, M is a p-group, and G has a single class of involutions. Furthermore, the number of groups for which H has a given positive number of regular orbits is finite.
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(PHD, 1964)

Abstract:

In this paper the relationship between the 2-length […] and the 2-exponent […] of a finite solvable group G is studied. It is shown that […] - 1 provided that […].

The special case of groups satisfying […] = 2, i.e., groups whose Sylow 2-groups are of exponent 4, is investigated to determine whether […] in this case. This question is not answered but it is shown that a certain normal subgroup (which may be the whole group) satisfies […]. In addition if all the elements of order 4 are contained in this subgroup, then […] for the whole group as well. As an application of this last result, it is proved that […] in a group of exponent 12
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(PHD, 1963)

Abstract:

This paper makes contributions to the structure theory of finite semifields, i.e., of finite nonassociative division algebras with unit. It is shown that a semifield may be conveniently represented as a 3-dimensional array of numbers, and that matrix multiplications applied to each of the three dimensions correspond to the concept of isotopy. The six permutations of three coordinates yield a new way to obtain projective planes from a given plane. Several new classes of semifields are constructed; in particular one class, called the binary semifields, provides an affirmative answer to the conjecture that there exist non-Desarguesian projective planes of all orders 2[…], if n is greater than 3. With the advent of binary semifields, the gap between necessary and sufficient conditions on the possible orders of semifields has disappeared.

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(PHD, 1962)

Abstract:

In this thesis finite groups whose maximal subgroups are of prime or prime square index are studied. The main problem considered is to find out to what extent this property is inherited by subgroups. The principal results are: this property is inherited by all subgroups if the group considered has odd order. This is not necessarily true if the group has even order. Let n be a positive integer. A group G of even order is constructed which contains a subgroup H, and H contains a maximal subgroup W with |H:W| larger than n.

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(PHD, 1962)

Abstract:

In this thesis primitive finite permutation groups G with regular abelian subgroup H are studied. It is shown that if, for an odd prime p, H has a Sylow p-subgroup which is the direct product of two cyclic groups of different order, then G is doubly transitive.

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