Special Relativity: are these postulates equivalent? At the beginning of the developing of special relativity the following principles are assumed true:

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*Principle 1: every physical law is equal in form in every inertial frame.

*Principle 2: there is a maximum finite velocity equal in all the inertial frames of reference that is equal to the speed of light.

*Definition: an inertial frame is a frame in which an object not
subjected to a force moves with uniform rectilinear motion.

However, in the developing of the theory it seems to me that a slightly different version of the first principle is used, namely:

*

*Principle $ \alpha$: the full symmetry group is given by translation in space and time, rotations and boosts of velocity. Namely, space and time are homogenous, space is isotropic and that laws of physics don't change after boosts of velocity.

To me it looks that principle $ \alpha$ can't be derived from principle 1 because principle 1 doesn't give any information about which are the transformation between an inertial frame to another
On the other side principle 1 can be derived from principle $ \alpha $  , for example:
from principle $ \alpha $ we know that after a boost the form of the physical laws doesn't change which implies that the dynamics remains unchanged. It follows that a uniform rectilinear motion will remain a uniform rectilinear motion and for this reason boosts transform an inertial frame in another inertial frame without changing the form of the equations.
The same can be said for translation in space and time and rotations.
So, question: is principle 1 equivalent to principle $ \alpha $ and consequently enogh to develop special relativity or do we need to restate it in a more complete form (that is principle $ \alpha $)
 A: The principle of relativity is a statement of symmetry. You here are interpreting it specifically as a symmetry under boosts, which is a rotation in spacetime. However, the usual interpretation in the literature is that it is a broader statement of symmetry including boosts, spatial rotations, and spacetime translations.
For example, Nothing But Relativity by Pal (Eur.J.Phys.24:315-319,2003) is a paper that explicitly uses homogeneity and isotropy as mathematical features included in the principle of relativity:
https://arxiv.org/abs/physics/0302045
So the usual interpretation of the principle of relativity is actually essentially the same as your reformulation.
A: Hypothesis 1 is not actually a hypothesis but a principle that helps select inertial frames. Without it, every frame is possible because a law of physics in one frame is also a law of physics in another frame - but they will look different.
Further, it's only the second sentence in Hypothesis $\alpha$ that is roughly equivalent to Hypothesis 1. And this is because it's merely a restatement of that hypothesis, if one additionally takes into account the full symmetry group.
The underlying hypothesis of isotropy and homogeniety is a hypothesis already made in classical mechanics and this did not change in relativity. By placing this hypothesis in with the hypothesis of the correct spacetime symmetry group only confuses matters rather than clarify.
