what is the definition of justice propose by thrasymachus? what isdd the argument used by socrates to refute this definition?

Thrasymachus' definition of justice: the interest of the stronger. Both terms of this definition are quickly brought into question, and, enraged, Thrasymachus unleashes a long diatribe, asserting that injustice benefits the ruler absolutely. Socrates, composed as ever, refutes him, offering true rule as just rule, for it is conducive to harmony, unity, and strength.

The dialogue concludes with Socrates' examination of the comparative advantages of justice and injustice. By the end, Thrasymachus and the other auditors are satisfied that the just man is happy, and the unjust is not. However, in a brilliant twist, Socrates dolefully admits to them that in spite of all the conversation, he still knows nothing about the nature of justice, but only something of its relation to virtue and not vice, wisdom and not ignorance, and of its utility over injustice. Presumably, the characters now return to the banquet from which they came, completing the circle.