Topology of the orthogonal group

Table of contents
- The orthogonal group is compact Topology of the orthogonal group
- Connected components of the orthogonal group Topology of the orthogonal group

The orthogonal group is compact

0 0

Connected components of the orthogonal group (The orthogonal group has two connected components)

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The orthogonal group has 2 connected components:

one with determinant +1, which is itself a subgroup known as the special orthogonal group. These are pure rotations without a reflection.
the other with determinant -1. This is not a subgroup as it does not contain the origin. It represents rotations with a reflection.

It is instructive to visualize how the

\pm 1

looks like in

S O (3)

you take the first basis vector and move it to any other. You have therefore two angular parameters.
you take the second one, and move it to be orthogonal to the first new vector. (you can choose a circle around the first new vector, and so you have another angular parameter.
at last, for the last one, there are only two choices that are orthogonal to both previous ones, one in each direction. It is this directio, relative to the others, that determines the "has a reflection or not" thing

As a result it is isomorphic to the direct product of the special orthogonal group by the cyclic group of order 2:

O (n) ≅ S O (n) \times C_{2}

(1)

A low dimensional example:

O (1) ≅ S O (2) \times C_{2}

(2)

because you can only do two things: to flip or not to flip the line around zero.

Note that having the determinant plus or minus 1 is not a definition: there are non-orthogonal groups with determinant plus or minus 1. This is just a property. E.g.: