Thesis (Ph.D.) - University of St Andrews, March 2007.

"The object of this paper is to present a number of theorems concerned with multiple transitivity in groups of permutations, culminating in a theorem of G. A. Miller on limits of transitivity of a group G in terms of the degree of G which is the number of letters on which the permutations of G act. The symmetric group consisting of all possible permutations on the n letters, is n - ply transitive. The alternating group, consisting of those permuations of the symmetric group which, when applied to the variables x₁,...,x[subscript n] carry the function [delta] = [pi] over i [lesser than] k (x[subscript i] - x[subscript k]) into itself, is (n-2) - ply transitive. In addition to the symmetric and alternating groups there are infinitely many groups which are 3 - ply transitive, but only a few groups known to be 4 - ply transitive. Using Miller's theorem it can be shown that for n [greater than] 12, a group of degree n cannot be t - fold transitive for t [less than or equal to] 3[square root of n]-2 unless the group is the symmetric or alternating group. Still better limits have been obtained since Miller published his theorem in 1915. Most recently, E. Parker obtained a limit with t of the order of magnitude 3[square root of n] for reasonable values of n”--Introduction. / "January, 1960." / Typescript. / "Submitted to the Graduate School of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Advisor: Nickolas Heerema, Professor Directing Paper. / Includes bibliographical references (leaf 27).

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Large scale comparative genetic mapping offers exciting prospects for understanding genomic evolution and has recently become of interest in computational molecular biology. The genome rearrangement problem is the computational problem of determining the smallest number of evolutionary events required to transform a given genome into another. In this thesis, we study a specific variant of the genome rearrangement problem. We assume that every genome has exactly one linear chromosome, and that each gene is an oriented unit and appears exactly once per genome. Furthermore, our model allows only two kinds of evolutionary events: reversals and transpositions. The problem is equivalent to the problem of sorting signed permutations by transpositions and reversals. We explore the characteristics of signed permutations and their sorting path. This exploration results in lower and upper bounds for a shortest sorting path. These bounds help us develop three approximation algorithms. We also prove that sorting by transpositions and reversals is at least as hard as the problem sorting by transpositions only - the complexity of which is unknown. In an effort to implement an algorithm to find an optimal sorting path of events, we designed four techniques to reduce the input size of the problem and thus achieve an improvement of the actual running time for any exhaustive algorithm.

Thesis (Ph. D.)--Rutgers University, 2008. / "Graduate Program in Mathematics." Includes bibliographical references (p. 107-109).

Errate slip inserted. Bibliography: leaves 130-134.

Thesis (Ph. D.)--New York University, 1935. / Planographed. Bibliography: p. 38-41.

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Thesis (Ph. D.)--University of Adelaide, 1988. / Errate slip inserted. Includes bibliographical references (leaves 130-134).