Connected components of a Lie group

From: Mark Adams (markjadams_at_lycos.com)
Date: 09/20/04 

Date: 20 Sep 2004 14:15:02 -0400

OK, one more dumb question on Lie groups: the identity component G_e of
any Lie group G is normal, and the quotient G/G_e is a finite group.
Since I never see G written as a semidirect product of G_e and G/G_e, I
assume there must be Lie groups where G/G_e is not isomorphic to a
finite subgroup of G. Can anyone provide an example of this?

I apologize for the series of naive questions, clearly this subject is
quite deep and should be studied in earnest at some point, which I hope
to do soon...