# Why do Lorentz boosts not form a group?

An excerpt from my lecture notes on relativity (translated from Dutch):

"Special (special in the notes indicates that the determinant of the representation matrix equals +1) Lorentz transformations with arbitrary velocities don't form a group. Special Lorentz transformations form a group (an Abelian subgroup of the Lorentzgroup) only when the boosts are parallel."

I don't see this. Why don't arbitrary boosts in arbitrary directions form a group? What criterion (closed under the operation, existence of the inverse, containing neutral element etc) of forming a group is not satisfied here?

• If by special you ean has determinant=1 then we have a group pf course, but I think the word "special" in regard to Lorentz tranformations means a "boost." Composing two boosts in non-parallel directions does not result in a a boost. – mike stone Jan 19 '20 at 21:59
• – Qmechanic Jan 19 '20 at 22:02
• Please edit this question so that it is clear whether you are talking about boosts or Lorentz transformations with determinant +1 when you say "special Lorentz transformation". Currently it's inconsistent. – ACuriousMind Jan 19 '20 at 22:10