Dada uma superfície M, definiremos os grupos de tranças de M, denotado por \'B IND. n\' (M), geometricamente e usando a noção de espaços de confiuração. Mostraremos a equivalência das definições. Na mesma linha de raciocínio, definiremos os grupos de tranças puras de superfícies \'P IND. n\' (M). Apresentaremos as propriedades mais importantes dos grupos de tranças do plano e mostraremos que \'B IND. n\' (\'R POT. 2\') injeta em \'B IND. n\' (M), para muitas superfícies M. Mais detalhadamente, obteremos a apresentação de \'B IND. n\' (\'RP POT. 2\' ) e \'P IND. n\'(\'RP POT. 2\') / For a surface M, we define the braid groups of M, \'B IND. n\'(M), geometricaly and using the notion of configuration spaces. We show the equivalence of these definitions. In the sequence, we define the pure braid group of M, \'P IND. n\' (M). We present the most important properties of braid groups of the plane and we show that \'B IND. n\'\'(\'R POT. 2\') embedds in \'B IND. n\' (M), for almost all M. In a more detailed fashion, we present \'B IND. n\' (\'RP POT. 2\') and \'P IND. n\' (\'RP POT. 2)

Computational group theory deals with the design, analysis and computer implementation of algorithms for solving computational problems involving groups, and with the applications of the programs produced to interesting questions in group theory, in other branches of mathematics, and in other areas of science. This thesis describes an implementation of a proposal for a Soluble Quotient Algorithm, i.e. a description of the algorithms used and a report on the findings of an empirical study of the behaviour of the programs, and gives an account of an application of the programs. The programs were used for the construction of soluble groups with interesting properties, e.g. for the construction of soluble groups of large derived length which seem to be candidates for groups having efficient presentations. New finite soluble groups of derived length six with trivial Schur multiplier and efficient presentations are described. The methods for finding efficient presentations proved to be only practicable for groups of moderate order. Therefore, for a given derived length soluble groups of small order are of interest. The minimal soluble groups of derived length less than or equal to six are classified.

Dada uma superfície M, definiremos os grupos de tranças de M, denotado por \'B IND. n\' (M), geometricamente e usando a noção de espaços de confiuração. Mostraremos a equivalência das definições. Na mesma linha de raciocínio, definiremos os grupos de tranças puras de superfícies \'P IND. n\' (M). Apresentaremos as propriedades mais importantes dos grupos de tranças do plano e mostraremos que \'B IND. n\' (\'R POT. 2\') injeta em \'B IND. n\' (M), para muitas superfícies M. Mais detalhadamente, obteremos a apresentação de \'B IND. n\' (\'RP POT. 2\' ) e \'P IND. n\'(\'RP POT. 2\') / For a surface M, we define the braid groups of M, \'B IND. n\'(M), geometricaly and using the notion of configuration spaces. We show the equivalence of these definitions. In the sequence, we define the pure braid group of M, \'P IND. n\' (M). We present the most important properties of braid groups of the plane and we show that \'B IND. n\'\'(\'R POT. 2\') embedds in \'B IND. n\' (M), for almost all M. In a more detailed fashion, we present \'B IND. n\' (\'RP POT. 2\') and \'P IND. n\' (\'RP POT. 2)