Hello, I consider the groups of rotations [itex]R=SO(2)[/itex] and the group [itex]T[/itex] of translations on the 2D Cartesian plane. Let's define Ω as the group Ω=RT. Thus Ω is essentially SE(2), the special Euclidean group. It is known that R and T are respectively 1-dimensional and 2-dimensional Lie groups diffeomorphic to the unit circle and the 2D plane. My question is: If I consider now [itex]\left\langle R \right\rangle ^\Omega[/itex], the conjugate closure of R with respect to Ω, what is the "structure" of such a group? Is it still a Lie group? if so, what is the manifold associated with it?