Lorentz group

It is the group of Lorentz transformations, and it is usually denoted by $O(3,1)$. It consist of usual rotations of ${\mathbb{R}}^3$ together with Lorentz boosts or hyperbolic rotations. It acts naturally on Minkowski space.
It is a subgroup of the Poincare group $Poin(3,1)$ in the same sense that $GL(3)$ is a subgroup of $\text{Aff}(3)$ and that $O(3)$ is a subgroup of $E(3)$.

While $O(n)$ has two connected components, $O(1,n)$ has four connected components. The connected component that contains the identity is $S{O}^{+}(1,n)$, and consists of those transformation that have determinant equal to 1 and map the vector $e_0$ (the time axis) to a future-pointing vector.