> 2) The Lorentz transformations form a group.

They don't "form" a group; they ARE a group, as named by their
inventor, Poincare' who called them "The Lorentz Transformation
Equatiom Group".
  Why a"Group"? Because there are four of them; one for x, one for y.
one for z and one for t, and four is a group.

glird