The pull string for the on/off switch pulled out of the socket. Can the pull string be put back into the socket?

Yes, you can, on some. Easier to replace.
If you do, remove from housing, Watch for spring wire, and parts. Probably less than 6 parts.
Usually chain breaks, Remove broken piece, install good chain.
If string, Resecure.

You have to replace the entire switch. See:

Ceiling fan pull chain replacement and repair - Ceiling Fans N More