The special orthogonal group 
is the subgroup of the orthogonal group 
consisting of orthogonal
matrices with determinant 1,
 |
(1)
|
Here 
is the 
identity matrix. Its elements are the orientation-preserving
linear isometries of Euclidean
space 
.
In particular, 
consists of the rotations of the plane,
while 
is the rotation group of three-dimensional space.
The group 
is a connected compact
Lie group of dimension 
. Its Lie algebra is
the space of real antisymmetric matrices,
 |
(2)
|
where 
denotes the set of all 
real matrices. The determinant
distinguishes the two components of 
and gives the exact sequence
 |
(3)
|
For 
,
the group center of 
is trivial when 
is odd and is 
when 
is even. The abelian group 
is its own group center. The special
unitary group 
is the universal cover
and double cover of 
.
More generally, 
denotes the isometries
of a nonsingular quadratic form 
having determinant 1. For
finite fields of odd characteristic,
this gives the subgroup 
of the general
orthogonal group 
. In characteristic
2, the determinant is identically 1 on the full orthogonal
group, so the determinant-one subgroup
is the full group.
, 
, 
, and 
, and 
." §2.4 in