Why is 'good friends' in the genitive case here? I thought that the genitive case was used for negative sentences, after certain prepositions and to indicate possession. In this example, what exactly do the 'good friends' possess that puts them into the genitive case?
Almost. "dobrą grupy" don't match each other grammatically, "mam" needs Accusative, which will be "dobrą grupę". Just like it's "grupę" in the main answer here.
wouldn't this sentence be the same even without the possession ("of")? If we take down the "grupa" from this sentence, wouldn't it still be "Mam dobrych przyjaciól" because of the mascular plural? (assuming there's at least 1 male among the friends?), so Acc=Gen?