I’ll keep this short and sweet:
(1) Book VII is the most famous portion of the Republic, because it begins with the “Allegory of the Cave.” So first: explain the allegory.
(2) What do the various people in the allegory symbolize? (Who are the prisoners? What are the shadows on the wall? Who are the people making the shadows? What does the fire represent? What is the meaning of the prisoner’s ascension and escape from the cave? What do the objects he encounters on the outside represent? In particular, what does the sun represent?).
(3) What is the allegory depicting? Is it life in the ideal city? Or is it life in a non-ideal city?
(4) What are some of the key aspects of the rulers’ education? Why do they need to spend so much time on mathematics?
(5) Why shouldn’t the guardians learn dialectics until the end of the process? How does this position relate to the one Socrates adopted in “The Apology”?