Well, "(grown) woman" is γυναίκα, but I translated κοπέλα as "young woman" and I agree that that translation should be acceptable. I reported it. (2/6/2017)
No, you'd need a possessive pronoun, just like in English you wouldn't say 'The girlfriend has become famous'.
I can't claim to know the intricacies of Greek and how the possessive might change the meaning, but in English at least, "The girlfriend has become famous" is a perfectly grammatical sentence.
Hmmm... I don't doubt that it is grammatically correct, but it does not seem to make much sense to me without the possessive. In Greek at least, unless it is someone's κοπέλα, it's only a young lady, not a girlfriend. Edit: And I just saw it in English, how it could work!
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