Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 8 / d / Solution

The subgroup generated by $T,P$ is $H=\{I,T,P,-I\}$, a Klein four-group. It is not normal in $G^{\prime }$. Indeed,For $x\ne 0$ the off-diagonal entries are nonzero, whereas every member of $H$ is diagonal. Thus this conjugate of $T$ is outside $H$, givingThe normal subgroup in part (c) is the continuous boost subgroup; choosing one representative from each of its cosets does not make those representatives a normal subgroup.