[[Lorentz group]]
# Proper Lorentz group

The **proper Lorentz group** $L_{+} = \mathrm{SO}(3,1)$ is the group of all [[Lorentz group|Lorentz transformations]] preserving the orientation of space, #m/def/group/lorentz
i.e.
$$
\begin{align*}
\mathrm{SO}(3,1) &= \{ \Lambda \in \mathrm{O}(3,1) : \det \Lambda = 1 \} 
\end{align*}
$$
where $\mathrm{O}(3,1)$ is the [[Lorentz group]].
As the [[Kernel of a group homomorphism|kernel]] of the map $\det : \mathrm{O}(3,1) \to \mathbb{Z}_{2}$ it is a [[normal subgroup]] of $\mathrm{O}(3,1)$.

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#state/develop | #lang/en | #SemBr