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What seems to be true but I don't understand why is that if a and b are elements in G, that if p(a) = p(b) then aK = bK in the factor group. Is this true? It seems like it should be. If so, then can somebody help me understand why it is true on an intuitive level?

Sorry, I realize that this is a broad and poorly phrased question, hoping somebody can see through my lack of competent communication and give me some insight here.