# Orthogonal group ($O(n)$) | ðŸ—– nosplit | â†‘ parent "Important Lie groups" | 154, 1, 175

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:

$O(n)â‰…SO(n)Ã—Z_{2}$

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

$O(1)â‰…Z_{(}2)$