Proper Lorentz group

The proper Lorentz group $𝐿 + = S O (3, 1)$ is the group of all Lorentz transformations preserving the orientation of space, lorentz i.e.

S O (3, 1) = {Λ \in O (3, 1) : d e t Λ = 1}

where $O (3, 1)$ is the Lorentz group. As the kernel of the map $d e t : O (3, 1) \to ℤ 2$ it is a normal subgroup of $O (3, 1)$ .