Thesis (Ph. D.)--University of Chicago, Department of Mathematics, June 2002. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Oregon, 2005. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 44-45). Also available for download via the World Wide Web; free to University of Oregon users.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1999. / Includes bibliographical references. Also available on the Internet.

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The purpose of this thesis was to present the technique of double coset enumeration and apply it to construct finite homomorphic images of infinite semidirect products. Several important homomorphic images include the classical groups, the Projective Special Linear group and the Derived Chevalley group were constructed.

This thesis constructs several groups entirely by hand via their symmetric presentations. In particular, the technique of double coset enumeration is used to manually construct J₃ : 2, the automorphism group of the Janko group J₃, and represent every element of the group as a permutation of PSL₂ (16) : 4, on 120 letters, followed by a word of length at most 3.

The purpose of the thesis is to give an alternative and more efficient method for working with finite groups by constructing finite groups as homomorphic images of progenitors. The method introduced can be applied to all finite groups that possess symmetric generating sets of involutions. Such groups include all finite non-abelian simple groups, which can then be constructed by the technique of manual double coset enumeration.

Advantages of the double coset enumeration technique include its use to represent group elements in a convenient shorter form than their usual permutation representations and to find nice permutation representations for groups. In this thesis we construct, by hand, several groups, including U₃(3) : 2, L₂(13), PGL₂(11), and PGL₂(7), represent their elements in the short form (symmetric representation) and produce their permutation representations.

Let X be a totally ordered set and consider the semigroups of orderdecreasing (increasing) full (partial, partial one-to-one) transformations of X. In this Thesis the study of order-increasing full (partial, partial one-to-one) transformations has been reduced to that of order-decreasing full (partial, partial one-to-one) transformations and the study of order-decreasing partial transformations to that of order-decreasing full transformations for both the finite and infinite cases. For the finite order-decreasing full (partial one-to-one) transformation semigroups, we obtain results analogous to Howie (1971) and Howie and McFadden (1990) concerning products of idempotents (quasi-idempotents), and concerning combinatorial and rank properties. By contrast with the semigroups of order-preserving transformations and the full transformation semigroup, the semigroups of orderdecreasing full (partial one-to-one) transformations and their Rees quotient semigroups are not regular. They are, however, abundant (type A) semigroups in the sense of Fountain (1982,1979). An explicit characterisation of the minimum semilattice congruence on the finite semigroups of order-decreasing transformations and their Rees quotient semigroups is obtained. If X is an infinite chain then the semigroup S of order-decreasing full transformations need not be abundant. A necessary and sufficient condition on X is obtained for S to be abundant. By contrast, for every chain X the semigroup of order-decreasing partial one-to-one transformations is type A. The ranks of the nilpotent subsemigroups of the finite semigroups of orderdecreasing full (partial one-to-one) transformations have been investigated.

This thesis studies two topics in sequence analysis. In the first part, we investigate the large deviations of the shape of the random RSK Young diagrams, associated with a random word of size n whose letters are independently drawn from an alphabet of size m=m(n). When the letters are drawn uniformly and when both n and m converge together to infinity, m not growing too fast with respect to n, the large deviations of the shape of the Young diagrams are shown to be the same as that of the spectrum of the traceless GUE. Since the length of the top row of the Young diagrams is the length of the longest (weakly) increasing subsequence of the random word, the corresponding large deviations follow. When the letters are drawn with non-uniform probability, a control of both highest probabilities will ensure that the length of the top row of the diagrams satisfies a large deviation principle. In either case, both speeds and rate functions are identified. To complete our study, non-asymptotic concentration bounds for the length of the top row of the diagrams, are obtained for both models. In the second part, we investigate the order of the r-th, 1<= r < +∞, central moment of the length of the longest common subsequence of two independent random words of size n whose letters are identically distributed and independently drawn from a finite alphabet. When all but one of the letters are drawn with small probabilities, which depend on the size of the alphabet, the r-th central moment is shown to be of order n^{r/2}. In particular, when r=2, we get the order of the variance of the longest common subsequence.