It can be deduced from the Burnside Basis Theorem that if G is a finite p-group with d(G)=r then given any generating set A for G there exists a subset of A of size r that generates G. We have denoted this property B. A group is said to have the basis property if all subgroups have property B. This thesis is a study into the nature of these two properties. Note all groups are finite unless stated otherwise. We begin this thesis by providing examples of groups with and without property B and several results on the structure of groups with property B, showing that under certain conditions property B is inherited by quotients. This culminates with a result which shows that groups with property B that can be expressed as direct products are exactly those arising from the Burnside Basis Theorem. We also seek to create a class of groups which have property B. We provide a method for constructing groups with property B and trivial Frattini subgroup using finite fields. We then classify all groups G where the quotient of G by the Frattini subgroup is isomorphic to this construction. We finally note that groups arising from this construction do not in general have the basis property. Finally we look at groups with the basis property. We prove that groups with the basis property are soluble and consist only of elements of prime-power order. We then exploit the classification of all such groups by Higman to provide a complete classification of groups with the basis property.

Submitted by Lúcia Brandão (lucia.elaine@live.com) on 2015-12-14T16:18:34Z No. of bitstreams: 1 Dissertação - Cleber Pereira.pdf: 842050 bytes, checksum: aad16346d388b6770589a7b27b0e4328 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2016-01-20T17:22:34Z (GMT) No. of bitstreams: 1 Dissertação - Cleber Pereira.pdf: 842050 bytes, checksum: aad16346d388b6770589a7b27b0e4328 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2016-01-20T17:23:12Z (GMT) No. of bitstreams: 1 Dissertação - Cleber Pereira.pdf: 842050 bytes, checksum: aad16346d388b6770589a7b27b0e4328 (MD5) / Made available in DSpace on 2016-01-20T17:23:12Z (GMT). No. of bitstreams: 1 Dissertação - Cleber Pereira.pdf: 842050 bytes, checksum: aad16346d388b6770589a7b27b0e4328 (MD5) Previous issue date: 2014-06-04 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / After describing some important classes of groups, we found that place nilpotência and local supersolubilidade are the property of PC- groups Farttini , dirty groups conjunction classes are polycyclic by infinity. In particular , we note that if G is a group then PC- G / ɸ ( g ) is locally nilpotent ( supersolúvel ) if , and only if G is locally nilpotent ( supersolúvel ) . / Depois de descrevermos algumas importantes classes de grupos, verificamos que local nilpotência e local supersolubilidade são propriedades de Farttini de PC-Grupos, grupos cujas classes de conjunção são policíclicas por infinito. Particularmente, verificamos que se G é um PC-Grupo então G/ɸ(g) é localmente nilpotente (supersolúvel) se, e somente se, G é localmente nilpotente (supersolúvel).