Michael is the music teacher's friend.
I don't know which is correct.
Michael is (the music teacher's) friend.
Michael is (the) (music teacher's friend.)
Think of it as Michael is a friend of the music teacher.
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AnonymousMichael is (the music teacher's) friend.
It's this one. The determiner (the) goes with the determiner phrase (the music teacher's). In these constructions the main noun (friend) does not have another determiner of its own, so the is added in a paraphrase with of.
a friend's brother ~ the brothe