Orthogonal group ( $O (n)$ ) | 🗖 nosplit | ↑ parent "Important Lie groups" | 154, 1, 175

🗖 nosplit | ⇓ toc | ↑ parent "Important Lie groups" | Wikipedia | 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.

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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) ≅ S O (n) \times Z_{2}

(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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Table of contents | 154, 1, 175
- 1. Special orthogonal group ( $S O (n)$ ) | 21 | 🔗 link | 🗖 nosplit | ↑ parent "Orthogonal group"

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Orthogonal group (O(n)) | 🗖 nosplit | ↑ parent "Important Lie groups" | 154, 1, 175

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Orthogonal group ( $O (n)$ ) | 🗖 nosplit | ↑ parent "Important Lie groups" | 154, 1, 175