The purpose of this thesis is to develop original symmetric presentations of finite non-abelian simple groups, particularly the sporadic simple groups. We have found original symmetric presentations for the Janko group J1, the Mathieu group M12, the Symplectic groups S(3,4) and S(4,5), a Lie type group Suz(8), and the automorphism group of the Unitary group U(3,5) as homomorphic images of the progenitors 2*60 : (2 x A5), 2*60 : A5, 2*56 : (23 : 7), and 2*28 : (PGL(2,7):2), respectively. We have also discovered the groups 24 : A5, 34 : S5, PSL(2,31), PSL(2,11), PSL(2,19), PSL(2,41), A8, 34 : S5, A52, 2• A52, 2 : A62, PSL(2,49), 28 : A5, PGL(2,19), PSL(2,71), 24 : A5, 24 : A6, PSL(2,7), 3 x PSL(3,4), 2• PSL(3,4), PSL(3,4), 2• (M12 : 2), 37:S7, 35 : S5, S6, 25 : S6, 35 : S6, 25 : S5, 24 : S6, and M12 as homomorphic images of the permutation progenitors 2*60 : (2 x A5), 2*60 : A5, 2*21 : (7: 3), 2*60 : (2 x A5), 2*120 : S5, and 2*144 : (32 : 24). We have given original proof of the 2*n Symmetric Presentation Theorem. In addition, we have also provided original proof for the Extension of the Factoring Lemma (involutory and non-involutory progenitors). We have constructed S5, PSL(2,7), and U(3,5):2 using the technique of double coset enumeration and by way of linear fractional mappings. Furthermore, we have given proofs of isomorphism types for 7 x 22, U(3,5):2, 2•(M12 : 2), and (4 x 2) :• 22.

A transitive decomposition of a graph is a partition of the arc set such that there exists a group of automorphisms of the graph which preserves and acts transitively on the partition. This turns out to be a very broad idea, with several striking connections with other areas of mathematics. In this thesis we first develop some general theory of transitive decompositions, and in particular we illustrate some of the more interesting connections with certain combinatorial and geometric structures. We then give complete, or nearly complete, structural characterisations of certain classes of transitive decompositions preserved by a group with a rank 3 action on vertices (such a group has exactly two orbits on ordered pairs of distinct vertices). The main classes of rank 3 groups we study (namely those which are imprimitive, or primitive of grid type) are derived in some way from 2-transitive groups (that is, groups which are transitive on ordered pairs of distinct vertices), and the results we achieve make use of the classification by Sibley in 2004 of transitive decompositions preserved by a 2-transitive group.

The main object of study of this work are orbits in so-called minimal sheets in irreducible representations of semisimple groups. Let $G$ be a semisimple group. The notion of sheets goes back to Dixmier: Given a $G$-module $V$, the union of all orbits in $V$ of a fixed dimension is a locally closed subset. Its irreducible components are called sheets of $V$. We call a sheet minimal if it contains an orbit in $V$ of minimal strictly positive dimension among all orbits in $V$. In Chapter I, some notation is fixed and some basic results are proved. In Chapter II, we describe minimal sheets in simple $G$-modules, and study $G$-stable deformations of orbits in minimal sheets by means of an invariant Hilbert scheme. Invariant Hilbert Schemes have been introduced by Alexeev and Brion in 2005. These are quasi-projective schemes representing functors of families of $G$-schemes with prescribed Hilbert function. The discussion in Chapter II is closely related to the work of Jansou in the following way: Choose once and for all a highest weight vector $v_\lambda \in V(\lambda)$ for each dominant weight $\lambda \in \Lambda^+$, and let $X_\lambda = \overline{G v_\lambda} \subset V(\lambda)$ be the closure of the orbit $G v_\lambda$ of $v_\lambda$ in $V(\lambda)$. In his thesis Jansou investigates $G$-stable deformations of $X_\lambda$ in $V(\lambda)$. If $h_\lambda$ denotes the Hilbert function of $X_\lambda$, then Jansou proves that the invariant Hilbert scheme $Hilb^G_{h_\lambda}(V(\lambda))$ is an affine space of dimension 0 or 1, depending on $G$ and $\lambda$. Furthermore, he gives a complete list of all pairs $(G,\lambda)$ such that $Hilb^G_{h_\lambda}(V(\lambda))$ is an affine line. In the sequel, we call these weights Jansou-weights. The orbit $Gv_\lambda$ is of minimal strictly positive dimension among all $G$-orbits in $V(\lambda)$. There exist other orbit of the same dimension as $Gv_\lambda$ in $V(\lambda)$ if and only if $\lambda$ is an integral multiple of a Jansou-weight. Here, we start with a general orbit $X$ of minimal strictly positive dimension in a fixed simple $G$-module $V(\lambda)$, and we study $G$-stable deformations of $X$. In particular, we conjecture that the invariant Hilbert scheme parametrizing the $G$-stable deformations of $X$ in the closure of the sheet of $X$ is an affine space of dimension either 0 or 1. This will stand in contrast to the fact that the invariant Hilbert scheme parametrizing the $G$-stable deformations of $X$ in $V(\lambda)$ can look much more complicated. This is the content of Chapter III, in which we will focus on the group $\SL_2$, and compute some corresponding invariant Hilbert schemes. In particular, we study deformations of orbits of the form $SL_2 \cdot x^{d/2}y^{d/2}$ in the space $k[x,y]_d = V(d)$ of binary forms of degree $d$. It turns out that easiest accessible case is when $d$ is a multiple of 4, and even in this case the corresponding invariant Hilbert scheme can become very complicated. This reflects the principle that even in `simple' cases for invariant Hilbert schemes all possible sort of `bad' things (different irreducible components, non-reduced points, singularities) occur. (This `bad' behavior is also encountered in the case of the classical Grothendieck Hilbert scheme parametrizing closed subschemes of projective space with a given Hilbert polynomial.) In Chapter III Classical Invariant Theory is often used, and some computations are computer-based. Finally, in Chapter IV we turn our attention to not necessarily simple modules. In the multiplicity-free case important work has been done by Bravi and Cupit-Foutou. We translate some of their results to the case of not necessarily multiplicity-free modules. This corrects a result by Alexeev and Brion. Chapter IV is independent from the preceding chapters.

Thesis (M.S.)--Brigham Young University. Dept of Mathematics, 2006. / Includes bibliographical references (p. 30).

This thesis presents a partition of the class of homomorphisms between groupoids of n-tuples in a system g = (G,&,@), where G = { a,b,c,d,e }is a set of five elements such that: 1) a is the &-identity and annihilates all elements under @; 2) b is the @-identity; 3) d absorbs all all elements except e under & and all elements except a and e under @; 4) e absorbs all elements under & and all elements except a under @; 5) & is a binary operation on G and is commutative in G; 6) @ is a binary operation on G and is left-distributive over & in G.Matrices over g were examined for characteristics which would determine different atomic properties of homomorphisms. A matrix operation @ was defined, which allowed the homomorphisms of groupoids of the form, (G(n) , &), to be modeled by a matrix equation. Using the atomic proper ties, a partition of the class of homomorphisms between groupoids was developed, and an example of an element in each of its disjoint subsets was presented. A listing of theorems was also derived.Ball State UniversityMuncie, IN 47306

The O'Nan-Scott Theorem classifies finite primitive permutation groups into one of five isomorphism classes. This theorem is very useful for answering questions about finite permutation groups since four out of the five isomorphism classes are well understood. The proof of this theorem currently relies upon the classification of the finite simple groups, as it requires a consequence of this classification, the Schreier Conjecture. After reviewing some needed group theoretic concepts, I give a detailed proof of the O'Nan-Scott Theorem. I then examine how the techniques of this proof have been applied to an open problem which asks whether every finite lattice can be embedded as an interval into the subgroup lattice of a finite group.

The O'Nan-Scott Theorem classifies finite primitive permutation groups into one of five isomorphism classes. This theorem is very useful for answering questions about finite permutation groups since four out of the five isomorphism classes are well understood. The proof of this theorem currently relies upon the classification of the finite simple groups, as it requires a consequence of this classification, the Schreier Conjecture. After reviewing some needed group theoretic concepts, I give a detailed proof of the O'Nan-Scott Theorem. I then examine how the techniques of this proof have been applied to an open problem which asks whether every finite lattice can be embedded as an interval into the subgroup lattice of a finite group.

Thesis (Ph. D.)--Rutgers University, 2009. / "Graduate Program in Mathematics." Includes bibliographical references (p. 86-88).

Thesis (Ph. D.)--University of Oregon, 2002. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 118-120). Also available for download via the World Wide Web; free to University of Oregon users.

Thesis (M.S.)--University of North Carolina at Wilmington, 2003. / Vita. Includes bibliographical references (leaf : [31]).