The premise of The Republic can best be summarized by the following passage:
“But to come now to the decision between our two kinds of life, if we separate the most completely just and the most completely unjust man, we shall be able to decide rightly, but if not, not. How, then, is this separation to be made? Thus: we must subtract nothing of his injustice from the unjust man or of his justice from the just, but assume the perfection of each in his own mode of conduct. In the first place, the unjust man must act as clever craftsmen do: a first-rate pilot or physician, for example, feels the difference between impossibilities and possibilities in his art and attempts the one and lets the others go; and then, too, if he does happen to trip, he is equal to correcting his error. Similarly, the unjust man who attempts injustice rightly must be supposed to escape detection if he is to be altogether unjust, and we must regard the man who is caught as a bungler. For the height of injustice is to seem just without being so. To the perfectly unjust man, then, we must assign perfect injustice and withhold nothing of it, but we must allow him, while committing the greatest wrongs, to have secured for himself the greatest reputation for justice; and if he does happen to trip, we must concede to him the power to correct his mistakes by his ability to speak persuasively if any of his misdeeds come to light, and when force is needed, to employ force by reason of his manly spirit and vigor and his provision of friends and money; and when we have set up an unjust man of this character, our theory must set the just man at his side—a simple and noble man, who, in the phrase of Aeschylus, does not wish to seem but be good. Then we must deprive him of the seeming. For if he is going to be thought just he will have honors and gifts because of that esteem. We cannot be sure in that case whether he is just for justice' sake or for the sake of the gifts and the honors. So we must strip him bare of everything but justice and make his state the opposite of his imagined counterpart. Though doing no wrong he must have the repute of the greatest injustice, so that he may be put to the test as regards justice through not softening because of ill repute and the consequences thereof. But let him hold on his course unchangeable even unto death, seeming all his life to be unjust though being just, that so, both men attaining to the limit, the one of injustice, the other of justice, we may pass judgement which of the two is the happier.” - Plato,
The Republic, Book 2, 360a - 361a
Plato describes two kinds of men. One completely unjust, but with the appearance of a just man. The other completely just, but with the appearance of an unjust man. He then makes the point that, when given the choice, one should always choose the just man over the unjust man.
This is the core of what The Republic is about; the role of justice in the soul of man. The city-state is used as part of an
analogy to make his point, as has been shown in the passage I quoted previously.
But I am not here to lecture you on Plato. After all, you profess to have sufficient understanding of his works. I obviously didn't come up with this on my own, and I could help you to some sources, but as they say 'pearls before swine'.