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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?