# Poincaré group is semi-product of translations and Lorentz-group

1. Oct 7, 2012

### aoner

Hi everybody,

Can somebody help me with the following proof.
Show that for the Poincaré group

$P=T\odot L$

Where $T$ is the group of translations and $L$ is the Lorentz group and $P$ is the semi-direct product of the two subgroups

I know the axioms for a semi-direct product in this case are
• $T$ is an invariant subgroup of $P$ while $L$ is any subgroup of $P$
• $T\cap L=\{E\}$, where $E$ is the identity of $P$
• For every element $S$ in $P$ we have an element ${{S}_{1}}$ of $T$ and an element ${{S}_{2}}$ of $L$ for which: $S={{S}_{1}}\circ {{S}_{2}}$

How do I proof these axioms?