How Is the Abelianization of a Lie Group Defined?

[mnb96]

Hi,

the abelianization of a group G is given by the quotient G/[G,G], where [G,G] is the commutator subgroup of G. When dealing with finite groups, the commutator subgroup is given by the (normal) subgroup generated by all the commutators of G.

If we consider instead the case of G being a Lie group, how do we abelianize it? In particular, do we define the commutator subgroup of a Lie group G in the standard way, as all the elements of G obtained by finite sequences of commutators and their inverses?

Thanks.

 

[fresh_42]

Yes. It is still a group and group operations are analytic.

 

[mnb96]

Alright, thanks.
As an exercise I tried to consider the group G=RT of roto-translations on the 2D plane (which is a non-abelian group) and if my calculations are correct, the commutator subgroup of G turns out to be exactly the subgroup T of translations. Interesting.
From this I shall probably deduce that the abelianization of the group of roto-translation is isomorphic to the subgroup R of rotations.

 

[lavinia]

Generally when you have a Lie subgroup, H, then the coset space G/H, is a smooth manifold and the action of G on G/H is itself smooth.