Orthogonal group ($O(n)$)

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:

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