I have no problem with Galilean transformations. The question is, what group do you need if you want the speed of light to be finite and the same for all observers. The Lorentz group is a good approximation, but is it good enough?

Thanks for reply. I just tell you we can take same procedure for both the transformations. For any direction of boost

[tex]L=R^{-1} L_zR[/tex]

where R is rotation so that rotated z axis is the direction of boost.