Abelian Lie group

In geometry, an abelian Lie group is a Lie group that is an abelian group.
A connected abelian real Lie group is isomorphic to formula_1. In particular, a connected abelian (real) compact Lie group is a torus; i.e., a Lie group isomorphic to formula_2. A connected complex Lie group that is a compact group is abelian and a connected compact complex Lie group is a complex torus; i.e., a quotient of formula_3 by a lattice.
Let "A" be a compact abelian Lie group with the identity component formula_4. If formula_5 is a cyclic group, then formula_6 is topologically cyclic; i.e., has an element that generates a dense subgroup. (In particular, a torus is topologically cyclic.)