I would like to repair this floor lamp. The piece in the picture has snapped but I’ve no idea what it’s called to get a new part. Can anyone help?
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Pendant lamp holder or Pendant socket (or perhaps Pendant lamp socket, you never can tell what mixture of words will work best when there's a choice.) That's a very typical stock design with the large threads to hold the shade. As Ed says, you replace the whole thing.
If you can't find one with threads on the back end (since pendant can mean "hanging by a cord" as well as just "designed to hang down", drop the "pendant" but then you have to sort out the ones with the large threads to hold the shade.