How is known that the Earth core is solid?
This is a good observation. The reason we can measure S-wave (transverse) propagation in the inner core is because P-waves can set up S-waves and vice versa. When an S-wave hits the mantle-outer core boundary at an angle it has a vibration component which is normal to the boundary and this sets up a P-wave in the liquid inner core. The opposite happens at the outer core-inner core boundary, where the P-wave at the boundary sets up an S-wave in the solid inner core. In this way, we can measure S-wave transmission through the inner core.