system
Given the following post, respond as if you're commenting on Reddit:
human
Do any of the Gallilean moons of Jupiter or the large moons of any other gas giant have stable geostationary orbits?

Writing a novella and I had the idea for space elevators at these moons, but I wanted to check the hard science first, and I can find very little to no research on this online.
gpt
Assuming you mean geostationary orbit around the moons themselves, not the gas giants:

All the large moons in the solar system are tidally locked to the planet they orbit.  That means their rotational period is the same as their orbital period.  In this case, it is only possible for a satellite to stay stationary with respect to the moon's surface if it is located at one of the Lagrange points of that moon.  

For any of these moons, a space elevator could be built from the moon's surface and passing through the moon-planet L1 or L2 point.  This would be much easier to build than a space elevator around Earth for example, because of the lower forces involved, although it would be longer in terms of distance.  In fact, there are semi-serious proposals to build a space elevator from the Earth's Moon to the Earth-Moon L1 point.