The Frattini subgroup of a Lie group and the topological rank of a Lie group

Abstract

We define the Frattini subgroup of a topological group G as the set of non-generators. We describe it completely for connected Lie groups, first for the case that G is solvable and then in general. As an application we determine the topological rank of G, that is the minimal number of elements needed to generate a dense subgroup of G.