Special orthogonal group:SO(3,R)

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Definition

This group, denoted $SO(3,\mathbb {R} )$, is the special orthogonal group for the standard dot product over the field of real numbers in three dimensions. Explicitly, it is given by:

$\{A\in GL(3,\mathbb {R} )\mid AA^{T}={\mbox{Identity matrix}},\det A=1\}$

It is also isomorphic to the projective special unitary group $PSU(2,\mathbb {C} )$ of degree two over the field of complex numbers, or equivalently, to the inner automorphism group of the group of unit quaternions.

Arithmetic functions

Function	Value	Similar groups	Explanation
dimension of an algebraic group	3		As $SO(n,\_),n=3:n(n-1)/2=3(3-1)/2=3$
dimension of a real Lie group	3		As $SO(n,\mathbb {R} ),n=3:n(n-1)/2=3(3-1)/2=3$

Subgroup structure

Further information: classification of finite subgroups of SO(3,R)