I've read that an electron beam consisting of electrons that are pretty densely packed manages to stay intact because any one electron experiences a magnetic force caused by the other electrons which somewhat balances the electric repulsion. Of course, that makes sense.
However, that is only true in the ground frame. When we shift over to the frame of any one electron, all electrons would be at rest, because they're all moving at the same velocity. So ideally in this frame, the only force between the electrons would be electrostatic repulsion. And that would mean there is no magnetic force keeping it together.
I've understood that Lorentz forces are supposed to stay the same regardless of the reference frame. However, this doesn't appear to be true, in this situation, since the electric force appears to have equal magnitude in both the frames, while the magnetic force exists only in the first. Can someone highlight where I've gone wrong?