What is the name for a three-carbon saturated alkyl group?
Depends on the point of attachment.
A prefix for a 3-carbon alkyl group would be "prop-". Depending on the point of attachment, the alkyl group itself would be named either "propyl" or "isopropyl". If you are attaching it via the first carbon of the chain, then it would be called propyl. If you are attaching it via the middle carbon, it would be called "isopropyl" or "2-propyl".
I hope that helps!