Re: Index 2 subgroups of Lie groups-

My favorite example of a Lie group containing multiple index-2 subgroups is the full homogeneous (3+1)-dimensional Lorentz group, the group of 4x4 matrices L = (Lij; i, j = 0, 1, 2, 3) which leave the quadratic form -x0^2 + x1^2 + x2^2 + x3^2 invariant.

It is easily proved that L00^2 >= 1 and (det L)^2 = 1.

With these relations in mind we have the proper orthochronous Lorentz group,
characterized by L00 >= 1 and det L = 1. It is the connected component containing the identity transformation.
No time reversal, no space reversal.
Index 4 in the full Lorentz group. Index 2 in the Lorentz subgroups mentioned below.
The full Lorentz group consists of four connected components.

With only L00 >= 1 one gets the orthochronous Lorentz group. No time reversal, only space reversal.
Index 2 in the full Lorentz group.

With only det L = 1 one gets the proper Lorentz group. No space reversal, only time reversal.
Index 2 in the full Lorentz group.

Generalized Lorentz groups O(m, n), leaving invariant the (m+n)-dimensional form -Sum over 1...m of xi^2 + Sum over m+1...m+n of xi^2, are a fertile hunting ground for index-2 subgroups.

To answer the original question by James: yes.
The determinant mapping M -> det (M) from a matrix group into the reals (or into the complex numbers, if the ground field is the complex numbers) is a group homomorphism and thus yields a normal subgroup and a factor group.
For all dimensions n we have an O(n) (two connected components: det = 1 and det = -1) which contains a normal subgroup SO(n) (det = 1).
BTW, an index-2 subgroup is always a normal subgroup.

Ciao: Johan E. Mebius


Christopher J. Henrich wrote:

>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.)
>
.

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