Special orthogonal group over reals

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Definition

For any natural number n, the special orthogonal group over reals of degree n, denoted SO(n,\R) or SO_n(\R), is defined as the following group:

\{ A \in GL(n,\R) \mid \det(A) = 1, AA^T = I \}

This can also be described as the group of linear transformations of \R^n that are orientation-preserving and also preserve the dot product.

Viewpoints

As an algebraic group

As a Lie group

As a topological group

Particular cases

Value of n Name of group Special comments
1 trivial group none
2 circle group the only nontrivial abelian case
3 special orthogonal group:SO(3,R) the smallest non-abelian case. Also, has S^3 (group of unit quaternions) as a double cover
4 special orthogonal group:SO(4,R)