Linear Transformation Groups:

• rotation: I = identity matrix is the identity element. det R=1

  Rotation is a member of the Special Orthogonal(n-dimensional) group. det R =+/- 1. SO(n) member of Lie group.

• (x1, x2, x3) -> (-x1, -x2, -x3) spatial inversion (parity). det R=-1
• (x1, x2, x3) -> (x1, x2, -x3) reflection. det R=-1