Книга: Rank of a group: Cartan Subgroup, Mathematics, Group Theory, Cardinality, Generating Set of a Group, Dimension, P-group, Affine Group, Frattini Subgroup, Free Abelian Group

High Quality Content by WIKIPEDIA articles! If G is a finitely generated group, then the rank of G is a nonnegative integer. The notion of rank of a group is a group-theoretic analog of the notion of dimension of a vector space. Indeed, for p-groups, the rank of the group P is the dimension of the vector space P/?(P), where ?(P) is the Frattini subgroup. The rank of a group is also often defined in such a way as to ensure subgroups have rank less than or equal to the whole group, which is automatically the case for dimensions of vector spaces, but not for groups such as affine groups.