I couldn't figure out the grammar rule here. I know it is possessive pronoun but I don't know the case (it seems nominative here)
Sie kauft die blaue Mütze für ihren Bruder.
Please tell me why it is ihren here?
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In general, you should be able to determine the case by searching for the preposition in some dictionary or even on Google.
Actually, knowing the case taken by a preposition is is not complicated at all if you have a good memory (or have practiced enough).
By the way, this wikibook has a decent lesson about prepositions and cases.