Intuitive definition: real group of rotations + reflections.

Mathematical definition: group of orthogonal matrices.

Has 2 disconnected: pieces: one with determinant +1 (which is is a subgroup known as the special orthogonal group) and the other with determinant -1.

If a reflection is done, the determinant is -1.

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

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

$O(n)≅SO(n)×Z_{2}$

A low dimensional example:
because you can only do two things: to flip or not to filp the line around zero.

$O(1)≅Z_{(}2)$

- Important Lie groups | 0, 978, 14
- Lie group | 233, 2k, 24
- Differential geometry | 12, 2k, 25
- Geometry | 0, 2k, 34
- Mathematics | 17, 13k, 336
- Ciro Santilli's Homepage | 262, 182k, 3k

- Special orthogonal group | 22
- Unitary group | 82, 155, 3