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theabelianizationof a group G is given by the quotient G/[G,G], where [G,G] is thecommutatorsubgroupof 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.

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# Abelianization of Lie groups

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