Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 302 / 3 / a / Solution

The Lorentz group consists of linear maps $x\mapsto \Lambda x$ satisfying ${\Lambda }^T\eta \Lambda =\eta$. The Poincare group consists of affine isometries $x\mapsto \Lambda x+a$ and has multiplicationThus it is the semidirect product ${\mathbb{R}}^{1,3}⋊O(1,3)$, with the Lorentz group as the subgroup fixing the spacetime origin.