Consider Lorentz transformations (in the standard configuration). It is easily demonstrated that the resultant of two successive Lorentz transformations, with velocities and , respectively, is equivalent to a Lorentz transformation with velocity . Lorentz transformations obviously satisfy the associative law. The unit element of the transformation group is just a Lorentz transformation with . Finally, the inverse of a Lorentz transformation with velocity is a transformation with velocity . We can use similar arguments to show that translations, rotations, parity inversions, and general Lorentz transformations (i.e., transformations between frames which are not in the standard configuration) also possess group properties.
If we think carefully, we can see that the group properties of the above mentioned transformations are a direct consequence of the relativity principle. Let us again consider Lorentz transformations. Suppose that we have three inertial frames , , and . According to , if we can get from to by a Lorentz transformation, and from to by a second Lorentz transformation, then it must always be possible to go directly from to by means of a third Lorentz transformation. Suppose, for the sake of argument, that we can find three frames for which this is not the case. In this situation, the frame could be distinguished from the frame because it is possible to make a direct Lorentz transformation from to the former frame, but not to the latter. This violates the relativity principle and, therefore, this situation can never arise. We can use a similar argument to demonstrate that a Lorentz transformation must possess an inverse. The associative law and the requirement that a unit element exists are trivially satisfied.