Each finite group G of elements l,x,y, ... can be partitioned into non-intersecting sets of elements which are conjugate if and only if they belong to the same set. These sets are called the conjugate classes of G, or simply the classes of G. [...]

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The group first defined by Grothendieck for his work on preschemes [2] can be generalized to arbitrary additive categories, and under certain conditions it can be made into a ring. The emphasis in this thesis is on algebraic results, but as shown in [3], the Grothendieck group has useful topological applications as well. [...]

Thesis (Ph. D.)--University of Chicago, Department of Mathematics, June 2003. / Includes bibliographical references. Also available on the Internet.

No description available.

Graduation date: 1997

Thesis (Ph. D.)--University of Alabama at Birmingham, 2007. / Additional advisors: Nikolai Chernov, S. S. Ravindran, Alan Sprague, Min Sun. Description based on contents viewed Feb. 6, 2008; title from title screen. Includes bibliographical references.

This thesis studies subsemigroups of groups from three perspectives: automatic structures, ordinary semigroup presentations, and Malcev presentaions. [A Malcev presentation is a presentation of a special type for a semigroup that can be embedded into a group. A group-embeddable semigroup is Malcev coherent if all of its finitely generated subsemigroups admit finite Malcev presentations.] The theory of synchronous and asynchronous automatic structures for semigroups is expounded, particularly for group-embeddable semigroups. In particular, automatic semigroups embeddable into groups are shown to inherit many of the pleasant geometric properties of automatic groups. It is proved that group- embeddable automatic semigroups admit finite Malcev presentations, and such presentations can be found effectively. An algorithm is exhibited to test whether an automatic semigroup is a free semigroup. Cancellativity of automatic semigroups is proved to be undecidable. Study is made of several classes of groups: virtually free groups; groups that satisfy semigroup laws (in particular [virtually] nilpotent and [virtually] abelian groups); polycyclic groups; free and direct products of certain groups; and one-relator groups. For each of these classes, the question of Malcev coherence is considered, together with the problems of whether finitely generated subsemigroups are finitely presented or automatic. This study yields closure and containment results regarding the class of Malcev coherent groups. The property of having a finite Malcev presentation is shown to be preserved under finite Rees index extensions and subsemigroups. Other concepts of index are also studied.

Uses the techniques of symmetric presentations to manipulate elements of large sporadic groups and to represent elements of these groups in much shorter forms than their corresponding permutation or matrix representation. Undertakes to develop a nested algorithm and a computer program to manipulate elements of large sporadic groups.

Benson spaces of higher order are introduced extending the idea of N. Krugljak and M. Milman, A distance between orbits that controls commutator estimates and invertibilty of operators, Advances in Mathematics 182 (2004), 78-123. The concept of Benson shift operators is introduced and a class of spaces equipped with these operators is considered. Commutator theorems of higher order on orbit spaces generated by a single element are proved for this class. It is shown that these results apply to the complex method of interpolation and to the real method of interpolation for the case q=1. Two new characterizations are presented of the domain space of the "derivation" operator in the context of orbital methods. Comparisons to the work of others are made, especially the unifying paper of M. Cwikel, N. Kalton, M. Milman and R. Rochberg, A United Theory of Commutator Estimates for a Class of Interpolation Methods, Advances in Mathematics 169 2002, 241-312. / by Eva, Kasprikova / Thesis (Ph.D.)--Florida Atlantic University, 2009. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2009. Mode of access: World Wide Web.