Proper Lorentz transformations from group theory?

[xxxyyy]

Hi,
I was looking at this derivation
https://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations#From_group_postulates
and I was wondering
1- where does the group structure come from? The principle of relativity? or viceversa? or what?
2- why only linear transformations? I remember someone told me (may be wrong) it was because of the homogeneity od spacetime, but I don't know why...
Thank you!

 

[PeroK]

The requirement that the set of Lorentz transformations forms a group comes from the basic properties that we expect of spacetime and inertial frames of reference.

The transformations must be linear, otherwise the transformation would depend on a particular reference point for $(t, x, y, z) = (0,0,0,0)$.

There's a derivation from homogeneity and isotropy here:

http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf

 

[vanhees71]

The group structure comes from the demand that transformations between different inertial reference frames should be invertible. The transformation from a frame to itself, i.e., doing in fact nothing, for sure is a symmetry and doing two transformations is also just a transformation from one inertial reference frame to another one.

The transformations should be linear because it should map any uniform motion of any point particle in one frame to such a motion in any other.

To get Minkowski space and the Poincare transformations you have to assume in addition also that any inertial observer describes his space as Euclidean.