Ciro Santilli
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# Orthogonal group () | 🗖 nosplit | ↑ parent "Important Lie groups" | 154, 1, 175

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Intuitive definition: real group of rotations + reflections.
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Mathematical definition: group of orthogonal matrices.
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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.
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If a reflection is done, the determinant is -1.
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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.
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As a result it isomorphic to the direct product of the special orthogonal group by the cyclic group of order 2: $$O(n)≅SO(n)×Z2​ (31)$$
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A low dimensional example: $$O(1)≅Z(​2) (32)$$ because you can only do two things: to flip or not to filp the line around zero.
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