Question on conjugation closure of subgroups

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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?
 

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fzero
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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?

Since ##\Omega## is abelian, it seems straightforward that [itex]\left\langle R \right\rangle ^\Omega =R[/itex]. If we were to consider a higher-dimensional Euclidean group, it seems that this continues to hold, since the result follows from ##T## being a normal subgroup of ##\Omega##.

A slightly more interesting example would be to consider a rank ##m## subgroup ##R_m## of a sufficiently large ##R_n## and study ## \left\langle R_m \right\rangle ^{\Omega_n} ##. It seems easy to convince yourself that this is just ##R_n## itself, though a formal proof would probably be illuminating.
 
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Since ##\Omega## is abelian...
Why is Ω supposed to be abelian?
For sure T and R are abelian groups, but I don't think Ω=RT is abelian, because R is not a normal subgroup of Ω. In general rt≠tr. Or am I missing something?
 
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fzero
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Why is Ω supposed to be abelian?
For sure T and R are abelian groups, but I don't think Ω=RT is abelian, because R is not a normal subgroup of Ω. In general rt≠tr. Or am I missing something?
No, you're correct. So let's work through it explicitly, like I should have done in the first place. A general element of ##\Omega## is

$$ g(a,b,\theta) = t_1 (a) t_2(b) r(\theta). $$

We are to consider ##g^{-1} r g##. A particular combination would be ## t_1(a)^{-1}r(\phi )t_1(a) = t_1 (a(\cos\phi-1)) t_2(a\sin\phi)##. I expect that considering the general case would lead to ##\langle R\rangle^\Omega = \Omega##.

Perhaps my more complicated example would lead to ##\langle R_m\rangle^{\Omega_n} = \Omega_m## when properly analyzed.
 
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Hi fzero,

thanks for your help. I think you are right when you suggest that [itex]\Omega = \left\langle R^\Omega \right\rangle[/itex], because [itex]\left\langle R^\Omega \right\rangle[/itex] is supposed to be a group anyway, and if it's neither T nor R, then I don't know what else it can be, besides Ω itself.

The thing that bothers me is that, I can't prove that [itex]T \subseteq \left\langle R^\Omega \right\rangle[/itex] from the definition of conjugate closure.

---EDIT:---
Maybe I got it: the conjugate closure contains also elements of the kind: [itex](t^{-1}rt)\,(\tau^{-1}r^{-1}\tau)[/itex] and since T is normal we have that [itex]r(t\tau^{-1})r^{-1}=t' \in T[/itex], thus the above quantity reduces to [itex]tt'\tau[/itex] which is an element of T.
 
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