the Lorentz group
/ LOR-ents /
Collect together every coordinate change that leaves the laws of relativity looking the same — all the boosts and all the spatial rotations — and you get a closed system: do one then another, and the result is again one of the same kind. That closed system is the Lorentz group, the symmetry group of Minkowski spacetime.
The Lorentz group is the set of all linear transformations Lambda that preserve the Minkowski metric, i.e. all Lambda satisfying Lambda^T eta Lambda = eta, equivalently all Lambda that leave every interval ds^2 invariant. It is denoted O(1,3). It has six independent parameters — three rotation angles and three boost velocities (its Lie algebra so(1,3) has three rotation generators J and three boost generators K). It splits into four disconnected pieces by two signs: the sign of the determinant (proper det = +1 versus improper det = -1) and whether it preserves the time direction (orthochronous or not). The piece continuously connected to the identity — proper orthochronous, SO+(1,3) — is generated by ordinary rotations and boosts.
Requiring physical laws to be invariant under the Lorentz group is the modern definition of 'relativistic'. Its representations classify particles: scalars, four-vectors, and (via its double cover SL(2,C)) spinors — which is how spin-1/2 enters relativistic quantum mechanics. Adding spacetime translations to the Lorentz group gives the full Poincare group, whose irreducible representations are labeled by mass and spin. Caveat: the discrete operations parity P (space inversion) and time reversal T live in the disconnected components; a law can be Lorentz invariant in the connected sense yet violate P or T (as the weak interaction does) — full O(1,3) invariance is a stronger requirement than proper-orthochronous invariance.
A spatial rotation and a boost are both elements of the Lorentz group; composing a boost along x with a boost along y is again in the group, but it equals a boost combined with a rotation — the reason the pure boosts do not by themselves form a subgroup.
Rotations and boosts together close into the six-parameter Lorentz group.
The Lorentz group is non-compact (boosts run to infinite rapidity), which is why it has no nontrivial finite-dimensional unitary representations — a technical fact underlying why relativistic quantum theory needs infinite-dimensional (field) representations.