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.

[pt] Os cinco grupos de Mathieu, M24;M23;M22;M12 e M11, compõem a primeira família de grupos esporádicos do Teorema de classificacão dos grupos simples finitos. Neste trabalho apresentaremos os grupos de Mathieu e alguns objetos relacionados a construcão deles como o Código de Golay e o Sistema de Steiner. Também, no decorrer do texto, surgiram espontaneamente alguns subgrupos dos grupos de Mathieu. / [en] The five Mathieu groups, M24;M23;M22;M12 and M11, form the first family of sporadic groups of Theorem classification of finite simple groups. In this paper we present the Mathieu groups and some objects related to building them as the Golay code and the Steiner system. Also, throughout the text, arose spontaneously some subgroups of groups of Mathieu.

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.

This thesis demonstrates an alternative, concise but informative, method for representing group elements, which will prove particularly useful for the sporadic groups. It explains the theory behind symmetric presentations, and describes the algorithm for working with elements represented in this manner.