Lorentz groupThe Lorentz group is a subgroup of the Poincaré group. It consists of all isometries of Minkowski space with a fixed point.
The Lorentz group is generated by rotations and Lorentz boosts. Mathematically, it is SO(3,1).
An orthochronous transformation is one which preserves the direction of time. A proper transformation is on which preserves spatial orientations. SO(3,1) refers to the proper, orthochronous subgroup, while O(3,1) refers to the entire group, with its four disconnected parts.
See also Lorentz transformation