Re: Index 2 subgroups of Lie groups

In article <dstvij$25i9$1@xxxxxxxxxxxxxxxxx>, James
<James545@xxxxxxxxx> wrote:

Hi,

I know SO(2) is an index 2 subgroup of O(2). Are there any other Lie groups
that have index 2 subgroups? Is SO(n) also an index 2 subgroup of O(n)? (I
know O(2) is generated by rotations and reflections, and SO(2) is generated
by rotations, which is where we get index 2, but is this true in general for
O(n) and SO(n)?)
What about other Lie groups with index 2 subgroups other than orthogonal
groups?

Thank you,

If G is any Lie group, the G x (Z/2Z) is a Lie group containing (a copy
of) G as a subgroup of index 2.

More generally, if G has a continuous automorphism of period 2, then
there is a semidirect product of G with (Z/2Z), in which a copy of G is
the normal subgroup.

Hmm... are there any other cases? (I thought this would be an easy
exercise with the answer "No"... but on second thought I'm not at all
sure.)

--
Chris Henrich
http://www.mathinteract.com
God just doesn't fit inside a single religion.
.

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