Which of the following is correct?
- Yesterday, I met one of my friends.
- Yesterday, I met one of my friend.
It's "one of my friends". The other option would be ungrammatical.
Think of it this way: If you didn't have several friends, you wouldn't need to be specifying that you only met one friend. "One of" is indicating a particular member out of a set with multiple members in it, so it's correct to use the plural form: "One of my friends." If you only have one friend total, then you can just say "Yesterday, I met my friend."
Similarly, when you turn it around thus:
then it's correct to use the singular verb form, because you're still only talking about one person. (Many people get confused because "friends", plural, is right next to the verb "is", but "One" is the true subject of the sentence.)
With the possible exception of set phrases, when you say one of [noun], [noun] is used in the plural form.
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