The Lorentz group is the group 
of time-preserving linear isometries
of Minkowski space 
with the Minkowski
metric
 |
(where the convention 
is used). It is also the group of isometries
of three-dimensional hyperbolic geometry.
It is time-preserving in the sense that the unit time vector 
is sent to another vector 
such that 
.
A consequence of the definition of the Lorentz group is that the full group of time-preserving isometries of Minkowski space 
is the group
direct product of the group of translations of 
(i.e., 
itself, with addition as the group operation), with
the Lorentz group, and that the full isometry group of the Minkowski 
is a group extension of 
by the product 
.
The Lorentz group is invariant under space rotations and Lorentz transformations.
